Sequences and Series

Let a & d are first term and common diff of an AP.

$\begin{aligned} & \mathrm{S}_{40}=\frac{40}{2}[2 \mathrm{a}+39 \mathrm{~d}]=1030 \quad\text{..... (1)}\\ & \mathrm{~S}_{12}=\frac{12}{2}[2 \mathrm{a}+11 \mathrm{~d}]=57 \quad\text{..... (2)} \end{aligned}$

by (1) & (2)

$\begin{aligned} & \mathrm{a}=-\frac{7}{2} \quad \mathrm{~d}=\frac{3}{2} \\ & \therefore \mathrm{~S}_{30}-\mathrm{S}_{10}=\frac{30}{2}[2 \mathrm{a}+29 \mathrm{~d}]-\frac{10}{2}[2 \mathrm{a}+9 \mathrm{~d}] \\ & =20 \mathrm{a}-390 \mathrm{~d} \\ & =515 \end{aligned}$

$\begin{aligned} & \text { Let } S=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\ldots \\ & \frac{1}{7} S=\frac{1}{7}(5)+\frac{1}{7^2}(5+\alpha)+\ldots \infty \end{aligned}$

$\begin{aligned} &\frac{6}{7}(S)=5+\frac{1}{7} \alpha\left(\frac{1}{1-\frac{1}{7}}\right)\\ &6=5+\frac{\alpha}{6} \Rightarrow \alpha=6 \end{aligned}$

To find the value of $ S_{2025} $, calculate the partial sum

$ S_n = \sum\limits_{n=1}^{2025} \frac{1}{n(n+1)} = \sum\limits_{n=1}^{2025} \left( \frac{1}{n} - \frac{1}{n+1} \right) $

This series is telescopic, meaning most terms cancel out with each other:

$ = \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{2025} - \frac{1}{2026} \right) $

Simplifying the expression, we find:

$ = \frac{1}{1} - \frac{1}{2026} = 1 - \frac{1}{2026} = \frac{2025}{2026} $

Now, evaluate $\sqrt{2026 \cdot S_{2025}}$:

$ \sqrt{2026 \cdot \frac{2025}{2026}} = \sqrt{2025} = 45 $

For the arithmetic progression with the first term $-p$ and common difference $p$, the sum of the first six terms is given by:

$ \frac{6}{2} [-2p + (6-1)p] = 3(5p) = 15p $

Given that:

$ 15p = 45 $

We find:

$ p = 3 $

To determine the absolute difference between the $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P., compute:

$ |A_{20} - A_{15}| = |(-p + 19p) - (-p + 14p)| $

Substituting the value of $p$:

$ = |18p - 13p| = |5p| = 5 \times 5 = 25 $

Thus, the absolute difference is $ \boxed{25} $.

The first term, $ a = 3 $

Common difference, $ d $

The formula for the sum of the first $ n $ terms of an A.P. is:

$ S_n = \frac{n}{2} [2a + (n-1)d] $

Given:

$ S_4 = \frac{1}{5}(S_8 - S_4) $

This implies:

$ 5S_4 = S_8 - S_4 \quad \Rightarrow \quad 6S_4 = S_8 $

Substituting the sum formulas:

$ 6 \cdot \frac{4}{2}[2 \times 3 + (4-1)d] = \frac{8}{2}[2 \times 3 + (8-1)d] $

Simplifying:

$ 6 \times 2 [6 + 3d] = 4 [6 + 7d] $

$ 12(6 + 3d) = 4(6 + 7d) $

$ 72 + 36d = 24 + 28d $

$ 36d - 28d = 24 - 72 $

$ 8d = -48 $

$ d = -6 $

Now, to find $ S_{20} $:

$ S_{20} = \frac{20}{2} [2 \times 3 + (20-1)(-6)] $

$ S_{20} = 10 [6 + 19 \times (-6)] $

$ S_{20} = 10 [6 - 114] $

$ S_{20} = 10 \times (-108) $

$ S_{20} = -1080 $

Thus, the sum of the first 20 terms is $-1080$.

$\begin{aligned} & a_1, a_2, a_3, \ldots \ldots, a_{2 k} \quad \rightarrow \mathrm{A.P.}\\ & \sum_{\mathrm{r}=1}^{\mathrm{k}} \mathrm{a}_{2 \mathrm{r}-1}=40, \sum_{\mathrm{r}=1}^{\mathrm{k}} \mathrm{a}_{2 \mathrm{r}}=55, \mathrm{a}_{2 \mathrm{k}}-\mathrm{a}_1=27 \\ & \frac{\mathrm{k}}{2}\left[2 \mathrm{a}_1+(\mathrm{k}-1) 2 \mathrm{~d}\right]=40, \frac{\mathrm{k}}{2}\left[2 \mathrm{a}_2+(\mathrm{k}-1) 2 \mathrm{~d}\right]=55, \\ & \mathrm{~d}=\frac{27}{2 \mathrm{k}-1} \\ & \mathrm{a}_1=\frac{40}{\mathrm{k}}-(\mathrm{k}-1) \mathrm{d}=\frac{55}{\mathrm{k}}-\mathrm{kd} \\ & \mathrm{~d}=\frac{15}{\mathrm{k}} \Rightarrow \frac{27}{2 \mathrm{k}-1}=\frac{15}{\mathrm{k}} \Rightarrow 9 \mathrm{k}=10 \mathrm{k}-5 \\ & \therefore \mathrm{k}=5 \end{aligned}$

First, let us denote the first term of the G.P. by $a_1 = A$ and the common ratio (which is $> 1$, since the G.P. is increasing) by $r$. Then the terms are:

$ a_1 = A,\quad a_2 = Ar,\quad a_3 = Ar^2,\quad a_4 = Ar^3,\quad a_5 = Ar^4,\quad a_6 = Ar^5,\;\dots $

We are given:

$a_1 \cdot a_5 = 28$, i.e.

$ A \cdot (A r^4) \;=\; A^2 r^4 \;=\; 28. \quad (1) $

$a_2 + a_4 = 29$, i.e.

$ Ar \;+\; A r^3 \;=\; A(r + r^3) \;=\; 29. \quad (2) $

We want to find $a_6 = A r^5$.

1. Solve for $r$

From $\text{(1)}$:

$ A^2 r^4 = 28 \quad\Longrightarrow\quad A^2 = \frac{28}{r^4}. $

From $\text{(2)}$:

$ A\,\bigl(r + r^3\bigr) = 29 \quad\Longrightarrow\quad A = \frac{29}{r + r^3}. $

Plug $A$ from $\text{(2)}$ into $\text{(1)}$. After some algebra (or by a systematic approach), one finds that

$ r^2 = 28 \quad\Longrightarrow\quad r = \sqrt{28} \;=\; 2\sqrt{7} $

(since $r>1$, we take the positive root).

2. Solve for $A$

From (2), using $r = 2\sqrt{7}$:

$ r + r^3 = 2\sqrt{7} + (2\sqrt{7})^3 = 2\sqrt{7} + 56\sqrt{7} = 58\sqrt{7}. $

Hence,

$ A = \frac{29}{\,58\sqrt{7}\,} = \frac{1}{2\sqrt{7}}. $

3. Find $a_6$

$ a_6 = A\,r^5 = \frac{1}{2\sqrt{7}}\;\times\;(2\sqrt{7})^5. $

Compute $(2\sqrt{7})^5$. First, $(2\sqrt{7})^2 = 28$; thus

$ (2\sqrt{7})^5 = (2\sqrt{7})^4 \cdot (2\sqrt{7}) = 784 \times (2\sqrt{7}) = 1568\sqrt{7}. $

Therefore,

$ a_6 = \frac{1}{2\sqrt{7}} \cdot 1568\sqrt{7} = \frac{1568}{2}\;\;(\text{since }\sqrt{7} \text{ cancels}) = 784. $

Answer: $a_6 = 784$.

Hence, the correct option is Option B.

$\begin{array}{ll} \sum_{n=0}^{\infty} a r^n=57 & \Rightarrow \frac{a}{1-r}=57 \quad \text{.... (i)}\\ \sum_{n=0}^{\infty} a^3 r^{3 n}=9747 & \Rightarrow \frac{a^3}{1-r^3}=9747 \quad \text{.... (ii)} \end{array}$

$\begin{aligned} & \frac{\left(1-r^3\right)}{(1-r)^3}=\frac{(57)^3}{9747}=19 \\ & \Rightarrow \quad \frac{(1-r)\left(1+r+r^2\right)}{(1-r)^3}=19 \\ & \Rightarrow \quad 18 r^2-39 r+18=0 \\ & \Rightarrow \quad r=\frac{2}{3}, \frac{3}{2} \text { (rejected) } \\ & \therefore \quad a=19 \\ & \quad a+18 r \\ & \quad=19+12=31 \end{aligned}$

$\frac{1}{1 \cdot(1+d)}+\frac{1}{(1+d)(1+2 d)}+\ldots+\frac{1}{(1+9 d)(1+10 d)}=5$

Multiply and divide by $d$

$\begin{aligned} & \frac{1}{d}\left[\frac{d}{1 \times(1+d)}+\frac{d}{(1+d)(1+2 d)}+\ldots+\frac{1}{(1+9 d)(1+10 d)}\right]=5 \\ & \frac{1}{d}\left[\left(1-\frac{1}{1+d}\right)+\left(\frac{1}{1+d}-\frac{1}{1+2 d}\right)+\ldots+\left(\frac{1}{1+9 d}-\frac{1}{1+10 d}\right)\right]=5 \\ & \frac{1}{d}\left[1-\frac{1}{1+10 d}\right]=5 \\ & \frac{1}{d}\left[\frac{1+10 d-1}{1+10 d}\right]=5 \end{aligned}$

$\begin{aligned} & \frac{10}{1+10 d}=5 \\ & 1+10 d=2 \\ & d=\frac{1}{10} \\ & 50 d=50 \times \frac{1}{10}=5 \end{aligned}$

Let's denote the first term of the geometric progression by $a$ and the common ratio by $r$. The terms of the geometric progression can be written as follows:

First term: $a$

Second term: $ar$

Third term: $(ar^2)$

Fourth term: $(ar^3)$

Fifth term: $(ar^4)$

Sixth term: $(ar^5)$

Eighth term: $(ar^7)$

We are given two key pieces of information:

1. The sum of the second and sixth terms is $\frac{70}{3}$:

$ar + ar^5 = \frac{70}{3}$

2. The product of the third and fifth terms is 49:

$(ar^2) \cdot (ar^4) = 49$

$a^2 r^6 = 49$

$a^2 = \frac{49}{r^6}$

$a = \frac{7}{r^3}$

Substituting $a = \frac{7}{r^3}$ into the first equation:

$\frac{7}{r^3} \cdot r + \frac{7}{r^3} \cdot r^5 = \frac{70}{3}$

$\frac{7r}{r^3} + \frac{7r^5}{r^3} = \frac{70}{3}$

$\frac{7}{r^2} + \frac{7r^2}{1} = \frac{70}{3}$

Let $x = r^2$. Then:

$\frac{7}{x} + 7x = \frac{70}{3}$

Multiply through by 3x to clear the denominator:

$21 + 21x^2 = 70x$

Rearrange into a standard quadratic equation:

$21x^2 - 70x + 21 = 0$

Divide by 7 to simplify:

$3x^2 - 10x + 3 = 0$

Solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 3$, $b = -10$, and $c = 3$. Thus:

$x = \frac{10 \pm \sqrt{100 - 36}}{6}$

$x = \frac{10 \pm \sqrt{64}}{6}$

$x = \frac{10 \pm 8}{6}$

$x = 3$ or $x = \frac{1}{3}$

Since $x = r^2$ and $r$ is positive, we get $r = \sqrt{3}$ or $r = \frac{1}{\sqrt{3}}$. We need to choose the value that results in positive, increasing terms:

If $r = \sqrt{3}$:

$a = \frac{7}{r^3} = \frac{7}{(\sqrt{3})^3} = \frac{7}{3\sqrt{3}} = \frac{7}{3} \cdot \frac{1}{\sqrt{3}} = \frac{7\sqrt{3}}{9}$

Now we can determine the sum of the 4th, 6th, and 8th terms:

The 4th term is: $ar^3 = \frac{7\sqrt{3}}{9} \cdot 3\sqrt{3} = 7$

The 6th term is: $ar^5 = \frac{7\sqrt{3}}{9} \cdot 9\sqrt{3} = 21$

The 8th term is: $ar^7 = \frac{7\sqrt{3}}{9} \cdot 27(\sqrt{3}) = 49$

Adding these together:

$(4th + 6th + 8th terms) = 7 + 21 + 63 = 91$

Therefore, the sum of the 4th, 6th, and 8th terms is 91.

Correct answer:

Option C: 91

$\triangle A B C$ is an equilateral triangle having side $=a$ unit

Now, perimeter $=$ sum of all sides $=3 a$

Area $=\frac{\sqrt{3}}{4} a^2$

Now, $\ln \triangle D E F, D E=\frac{a}{2}=E F=D F$

$\begin{aligned} & \text { Perimeter }=3 \times \frac{a}{2}=\frac{3 a}{2} \\ & \text { Area }=\frac{\sqrt{3}}{4} \times\left(\frac{a}{2}\right)^2=\frac{\sqrt{3} a^2}{16} \end{aligned}$

Now, $P=3 a+\frac{3 a}{2}+\frac{3 a}{4}+\cdots$

$Q=\frac{\sqrt{3}}{4} a^2+\frac{\sqrt{3}}{16} a^2+\frac{\sqrt{3}}{64} a^2+\cdots$

$P=\frac{3 a}{1-\frac{1}{2}}=3 a \times 2 \Rightarrow P=6 a \quad \text{.... (i)}$

$Q=\frac{\frac{\sqrt{3}}{4} a^2}{1-\frac{1}{4}}=\frac{4}{3} \times \frac{\sqrt{3}}{4} a^2 \quad Q=\frac{\sqrt{3}}{3} a^2 \quad \text{.... (ii)}$

From equation (i) & (ii)

$\begin{aligned} & P=6 a \\ & Q=\frac{\sqrt{3}}{3} a^2 \\ & Q=\frac{\sqrt{3}}{3} \times \frac{P^2}{36} \\ & P^2=36 \sqrt{3} Q \end{aligned}$

To determine the value of $\mathrm{m}$, we need to formulate the problem using some basic concepts of arithmetic progression and work. Let's first understand the nature of the problem:

Initially, there are $\mathrm{m}$ computers, and it is estimated that with these $\mathrm{m}$ computers, the assignment can be completed in 17 days.

However, due to the crash of 4 computers every day starting from the second day onward, the total time taken extends by 8 days, making it 25 days in total.

To begin with, let's define the total work (W) in terms of the number of computers and days:

The total work (W) is given by:

The amount of work completed each day with $\mathrm{m}$ computers for 17 days:

$W = 17m$

When computers crash, the number of working computers each day forms an arithmetic sequence. On the first day, there are $\mathrm{m}$ computers. On the second day, there are $\mathrm{m} - 4$ computers, on the third day, there are $\mathrm{m} - 8$ computers, and so on. We need to sum this series until 25 days are completed.

This can be formulated as:

Total work done over 25 days with decrement in the number of computers:

$W = m + (m - 4) + (m - 8) + \ldots + \left[m - 4 \times (n - 1)\right]$

where $n$ is the number of days. Here, $n = 25$.

Notice that we form an arithmetic series where the first term (a) is $\mathrm{m}$ and the common difference (d) is -4. The sum of the first n terms of an arithmetic series is:

$S_n = \frac{n}{2} \left[ 2a + (n - 1)d \right]$

Plugging in the values:

$S_{25} = \frac{25}{2} \left[ 2m + (25 - 1)(-4) \right]$

$S_{25} = \frac{25}{2} \left[ 2m - 96 \right]$

$S_{25} = \frac{25}{2} \left[ 2m - 96 \right] = 25(m - 48)$

This work should be equivalent to the work calculated earlier, so:

$17m = 25(m - 48)$

Solving for $\mathrm{m}$:

$17m = 25m - 1200$

$8m = 1200$

$m = 150$

Thus, the value of $\mathrm{m}$ is equal to:

Option C: 150

To determine the least value of $\mathrm{K}$ for which the terms $4^{1+x} + 4^{1-x}, \frac{\mathrm{K}}{2}, 16^x + 16^{-x}$ form an arithmetic progression (A.P.), we need to establish the relationship among these terms in an A.P.

For three numbers to be in an arithmetic progression, the middle term must be the average of the other two terms. Therefore, we can write:

$ \frac{4^{1+x} + 4^{1-x} + 16^x + 16^{-x}}{2} = \frac{K}{2} $

First, simplify each term individually:

1. Consider $4^{1+x} + 4^{1-x}$:

$4^{1+x} = 4 \cdot 4^x = 4 \cdot (2^2)^x = 4 \cdot 2^{2x} = 4 \cdot 2^{2x}$

and

$4^{1-x} = 4 \cdot 4^{-x} = 4 \cdot (2^2)^{-x} = 4 \cdot 2^{-2x} = 4 \cdot 2^{-2x}$

Thus,

$4^{1+x} + 4^{1-x} = 4 \cdot 2^{2x} + 4 \cdot 2^{-2x} = 4(2^{2x} + 2^{-2x})$

2. Consider $16^x + 16^{-x}$:

$16^x = (2^4)^x = 2^{4x}$

and

$16^{-x} = (2^4)^{-x} = 2^{-4x}$

Thus,

$16^x + 16^{-x} = 2^{4x} + 2^{-4x}$

3. Combine the terms and set up the equation:

$ \frac{4(2^{2x} + 2^{-2x}) + 2^{4x} + 2^{-4x}}{2} = \frac{K}{2} $

Multiply both sides by 2:

$ 4(2^{2x} + 2^{-2x}) + 2^{4x} + 2^{-4x} = K $

To find the least value of $\mathrm{K}$, let's assume $x = 0$ (since $x$ can range over non-negative values):

For $x = 0$:

$4(2^{2 \cdot 0} + 2^{-2 \cdot 0}) + 2^{4 \cdot 0} + 2^{-4 \cdot 0}$

This simplifies to:

$4(2^0 + 2^0) + 2^0 + 2^0$

$ = 4(1 + 1) + 1 + 1 $

$ = 4 \cdot 2 + 1 + 1 $

$ = 8 + 1 + 1 $

$ = 10 $

Therefore, the least value of $\mathrm{K}$ that ensures the values form an arithmetic progression is $ 10 $. Hence, the correct option is:

Option A: 10

$\begin{aligned} & \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m \\ & \text { and } \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=n \\ & \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}} \\ & =\frac{1}{\sqrt{1}+\sqrt{2}} \times \frac{\sqrt{2}-\sqrt{1}}{\sqrt{2}-\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} \\ & \quad+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}} \times \frac{\sqrt{100}-\sqrt{99}}{\sqrt{100}-\sqrt{99}} \end{aligned}$

$\begin{aligned} & =\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\ldots+\sqrt{100}-\sqrt{99} \\ & =\sqrt{100}-\sqrt{1} \\ & =10-1 \\ & \Rightarrow m=9 \end{aligned}$

and $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=n$

$\begin{aligned} & \frac{2-1}{1 \times 2}+\frac{3-2}{2 \times 3}+\ldots+\frac{100-99}{100 \times 99}=n \\ & \Rightarrow 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\ldots+\frac{1}{99}-\frac{1}{100}=n \\ & \Rightarrow n=1-\frac{1}{100} \\ & \Rightarrow n=\frac{99}{100} \\ & (m, n)=\left(9, \frac{99}{100}\right) \end{aligned}$

Satisfies the line $11 x-100 y=0$

$\begin{aligned} & \frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots+100^2 \times 101} \\ & \Rightarrow \frac{\sum_\limits{n=1}^{100} n(n+1)^2}{\sum_\limits{n=1}^{100} n^2(n+1)} \end{aligned}$

$\begin{aligned} & \Rightarrow \frac{\sum_\limits{n=1}^{100} n^3+2 n^2+n}{\sum_\limits{n=1}^{100} n^3+n^2} \\ & =\frac{\left(\frac{100(101)}{2}\right)^2+\frac{2 \cdot 100(101)(201)}{6}+\frac{100(101)}{2}}{\left(\frac{100(101)}{2}\right)^2+\frac{100(101)(201)}{6}} \\ & =\frac{300(101)+4(201)+6}{300(101)+2(201)}=\frac{5185}{5117}=\frac{305}{301} \end{aligned}$

$\begin{aligned} & 2 b=a+c \quad \text{.... (1)}\\ & b^2=(a+1)(c+3) \quad \text{.... (2)}\\ & \frac{a+b+c}{3}=8 \quad \text{.... (3)} \end{aligned}$

$\begin{aligned} \Rightarrow & \frac{3 b}{3}=8 \\ & b=8 \\ \Rightarrow \quad & a c+3 a+c+3=64 \end{aligned}$

$\begin{aligned} & 3 a+c+a c=61 \quad \text{... (4)}\\ & a+c=16 \\ & c=16-a \end{aligned}$

from equation (4)

$\begin{aligned} & 3 a+16-a+a(16-a)=61 \\ & \Rightarrow \quad(a-15)(a-3)=0 \\ & \quad a=15(a>10) \\ & \Rightarrow \quad a=15, b=8, c=1 \\ & \left((a \cdot b \cdot c)^{\frac{1}{3}}\right)^3=15 \times 8 \times 1=120 \end{aligned}$

$\begin{aligned} & \text { Let } p=2 r, q=2 r^2 \\ & T_7=2, T_8=2 r, T_{13}=2 r^2 \\ & d=2 r-2=2(r-1) \\ & 2 r^2=T_7+6 d=2+6(2)(r-1)=12 r-10 \\ & \Rightarrow r^2-6 r+5=0 \\ & \Rightarrow(r-1)(r-5)=0 \\ & \therefore r=1,5 \\ & r=1 \text { (rejected) as } q \neq 2 \\ & \therefore r=5 \end{aligned}$

$5^{\text {th }}$ term of G.P $=2 . r^4=2.5^4$

Let $1^{\text {st }}$ term of A.P $b a=a, d=8$

$2=a+(6)(8) \Rightarrow a=-46$

$\mathrm{n}^{\text {th }}$ term of A.P $=-46+(n-1) 8=8 n-54$

$\begin{aligned} & 2.5^4=8 n-54 \\ & \Rightarrow 1250+54=8 n \\ & \Rightarrow n=\frac{1304}{8}=163 \end{aligned}$

To solve this problem, we will start by using the properties of an arithmetic progression (AP).

The sum of the first $n$ terms of an AP can be calculated using the formula: $ S_n = \frac{n}{2} (2a + (n-1)d) $ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, and $d$ is the common difference between the terms.

Given the information: $ S_{10} = 390 $

We can plug $n=10$ into the sum formula to get:

$ S_{10} = \frac{10}{2} (2a + (10-1)d) $

$ 390 = 5(2a + 9d) $

$ 390 = 10a + 45d $

$ 78 = 2a + 9d \quad .........\text{(1)} $

Next, we're given the ratio of the tenth term ($T_{10}$) to the fifth term ($T_5$): $ \frac{T_{10}}{T_5} = \frac{15}{7} $

The $n$th term of an AP is given by:

$ T_n = a + (n-1)d $

So, for the tenth term: $ T_{10} = a + (10-1)d = a + 9d $

And for the fifth term:

$ T_5 = a + (5-1)d = a + 4d $

Now we can write the ratio as:

$ \frac{a + 9d}{a + 4d} = \frac{15}{7} $

$ 7(a + 9d) = 15(a + 4d) $

$ 7a + 63d = 15a + 60d $

$ 63d - 60d = 15a - 7a $

$ 3d = 8a \quad .........\text{(2)} $

Now we have two equations (1) and (2):

$ 78 = 2a + 9d \quad \text{(1)} $

$ 3d = 8a \quad \text{(2)} $

We can solve these equations simultaneously.

From equation (2):

$ d = \frac{8}{3}a $

Plugging this back into (1):

$ 78 = 2a + 9\left(\frac{8}{3}a\right) $

$ 78 = 2a + 24a $

$ 78 = 26a $

$ a = 3 $

Now we can find $d$:

$ d = \frac{8}{3}a $

$ d = \frac{8}{3} \times 3 $

$ d = 8 $

Now we can find $S_{15}$ and $S_5$ using the formula for the sum of an AP.

For $S_{15}$:

$ S_{15} = \frac{15}{2} (2 \cdot 3 + (15-1) \cdot 8) $

$ S_{15} = \frac{15}{2} (6 + 14 \cdot 8) $

$ S_{15} = \frac{15}{2} (6 + 112) $

$ S_{15} = \frac{15}{2} \cdot 118 $

$ S_{15} = 15 \cdot 59 $

$ S_{15} = 885 $

For $S_5$:

$ S_5 = \frac{5}{2} (2 \cdot 3 + (5-1) \cdot 8) $

$ S_5 = \frac{5}{2} (6 + 4 \cdot 8) $

$ S_5 = \frac{5}{2} (6 + 32) $

$ S_5 = \frac{5}{2} \cdot 38 $

$ S_5 = 5 \cdot 19 $

$ S_5 = 95 $

The difference $S_{15} - S_{5}$ is:

$ S_{15} - S_{5} = 885 - 95 $

$ S_{15} - S_{5} = 790 $

Therefore, the correct answer is Option C, which is 790.

Since $3, a, b, c$ are in arithmetic progression (A.P.), the common difference can be calculated using the term $a$ (the second term) as follows:

$ d = a - 3 $

The nth term of an A.P. is given by the formula:

$ T_n = a + (n-1)d $

So, using this formula, we can express $b$ and $c$ in terms of $a$ and $d$:

$ b = a + d $

$ c = a + 2d $

Substituting $d = a - 3$ into these expressions:

$ b = a + (a - 3) $

$ c = a + 2(a - 3) $

Therefore:

$ b = 2a - 3 $

$ c = 3a - 6 $

Now, let's consider that $3, a-1, b+1, c+9$ are in geometric progression (G.P.). For terms in a G.P., the ratio (common ratio, r) between consecutive terms is constant. So:

$ \frac{a - 1}{3} = \frac{b + 1}{a - 1} = \frac{c + 9}{b + 1} $

Now, we will establish the relation between the terms using the property of G.P.:

$ \frac{a - 1}{3} = \frac{b + 1}{a - 1} $

$ (a - 1)^2 = 3(b + 1) $

$ a^2 - 2a + 1 = 3b + 3 $

Substituting $b = 2a - 3$, we get:

$ a^2 - 2a + 1 = 3(2a - 3) + 3 $

$ a^2 - 2a + 1 = 6a - 9 + 3 $

$ a^2 - 8a + 7 = 0 $

Solving this quadratic equation:

$ (a - 7)(a - 1) = 0 $

Hence, $a = 7$ or $a = 1$. However, if $a = 1$, the terms $3, a-1, b+1, c+9$ cannot form a G.P. as it would involve division by zero. Therefore, $a = 7$. We use this value to find $b$ and $c$:

$ b = 2a - 3 = 2(7) - 3 = 14 - 3 = 11 $

$ c = 3a - 6 = 3(7) - 6 = 21 - 6 = 15 $

Now we can find the arithmetic mean ($A$) of $a$, $b$, and $c$:

$ A = \frac{a + b + c}{3} $

$ A = \frac{7 + 11 + 15}{3} $

$ A = \frac{33}{3} $

$ A = 11 $

Hence, the arithmetic mean of $a$, $b$, and $c$ is $11$, which corresponds to Option D.

$\begin{aligned} & 1+d, \quad 1+7 d, 1+43 d \text { are in GP } \\ & (1+7 d)^2=(1+d)(1+43 d) \\ & 1+49 d^2+14 d=1+44 d+43 d^2 \\ & 6 d^2-30 d=0 \\ & d=5 \\ & S_{20}=\frac{20}{2}[2 \times 1+(20-1) \times 5] \\ & \quad=10[2+95] \\ & \quad=970 \end{aligned}$

$\begin{aligned} & f(x)=(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b) \\ & f(x)=a+b-2 c+b+c-2 a+c+a-2 b=0 \\ & f(1)=0 \\ & \therefore \alpha \cdot 1=\frac{c+a-2 b}{a+b-2 c} \\ & \alpha=\frac{c+a-2 b}{a+b-2 c} \\ & \text { If, }-1<\alpha<0 \\ & -1<\frac{c+a-2 b}{a+b-2 c}<0 \\ & b+c<2 a \text { and } b>\frac{a+c}{2} \end{aligned}$

therefore, b cannot be G.M. between a and c.

$\begin{aligned} & \text { If, } 0<\alpha<1 \\ & 0<\frac{c+a-2 b}{a+b-2 c}<1 \\ & b>c \text { and } b<\frac{a+c}{2} \end{aligned}$

Therefore, $\mathrm{b}$ may be the G.M. between $\mathrm{a}$ and $\mathrm{c}$.

General term of the sequence,

$\begin{aligned} & \mathrm{T}_{\mathrm{r}}=\frac{\mathrm{r}}{1-3 \mathrm{r}^2+\mathrm{r}^4} \\ & \mathrm{~T}_{\mathrm{r}}=\frac{\mathrm{r}}{\mathrm{r}^4-2 \mathrm{r}^2+1-\mathrm{r}^2} \\ & \mathrm{~T}_{\mathrm{r}}=\frac{\mathrm{r}}{\left(\mathrm{r}^2-1\right)^2-\mathrm{r}^2} \\ & \mathrm{~T}_{\mathrm{r}}=\frac{\mathrm{r}}{\left(\mathrm{r}^2-\mathrm{r}-1\right)\left(\mathrm{r}^2+\mathrm{r}-1\right)} \\ & \mathrm{T}_{\mathrm{r}}=\frac{\frac{1}{2}\left[\left(\mathrm{r}^2+\mathrm{r}-1\right)-\left(\mathrm{r}^2-\mathrm{r}-1\right)\right]}{\left(\mathrm{r}^2-\mathrm{r}-1\right)\left(\mathrm{r}^2+\mathrm{r}-1\right)} \\ & =\frac{1}{2}\left[\frac{1}{\mathrm{r}^2-\mathrm{r}-1}-\frac{1}{\mathrm{r}^2+\mathrm{r}-1}\right] \end{aligned}$

Sum of 10 terms,

$\sum_\limits{\mathrm{r}=1}^{10} \mathrm{~T}_{\mathrm{r}}=\frac{1}{2}\left[\frac{1}{-1}-\frac{1}{109}\right]=\frac{-55}{109}$

The problem involves finding a relation between terms of two different geometric progressions (GPs) which share common first terms but have different terms equated to the same value. We solve this by setting up equations based on the given conditions for each GP and comparing the terms specified to be equal.

For the first GP, with first term $a$ and third term $b$:

• The third term is given by $t_3 = ar^2 = b$, leading to $r^2 = \frac{b}{a}$.
• The eleventh term is $t_{11} = ar^{10} = a\left(\frac{b}{a}\right)^5$.

For the second GP, with first term $a$ and fifth term $b$:

• The fifth term is $T_5 = ar^4 = b$, yielding $r^4 = \frac{b}{a}$ and $r = \left(\frac{b}{a}\right)^{1/4}$.
• The $p^{\text{th}}$ term is thus $T_p = ar^{p-1} = a\left(\frac{b}{a}\right)^{\frac{p-1}{4}}$.

Equating the eleventh term of the first GP to the $p^{\text{th}}$ term of the second GP gives: $a\left(\frac{b}{a}\right)^5 = a\left(\frac{b}{a}\right)^{\frac{p-1}{4}}$.

Simplifying, we find that $5 = \frac{p-1}{4}$, leading to $p = 21$, which is the solution.

$\begin{aligned} &\begin{aligned} & \mathrm{S}_{20}=\frac{20}{2}[2 \mathrm{a}+19 \mathrm{~d}]=790 \\ & 2 \mathrm{a}+19 \mathrm{~d}=79 \quad \text{.... (1)}\\ & \mathrm{~S}_{10}=\frac{10}{2}[2 \mathrm{a}+9 \mathrm{~d}]=145 \\ & 2 \mathrm{a}+9 \mathrm{~d}=29 \quad \text{.... (2)} \end{aligned}\\ &\text { From (1) and (2) } a=-8, d=5 \end{aligned}$

$\begin{aligned} & S_{15}-S_5=\frac{15}{2}[2 a+14 d]-\frac{5}{2}[2 a+4 d] \\ & =\frac{15}{2}[-16+70]-\frac{5}{2}[-16+20] \\ & =405-10 \\ & =395 \end{aligned}$

$\log _{\mathrm{e}} \mathrm{a}, \log _{\mathrm{e}} \mathrm{b}, \log _{\mathrm{e}} \mathrm{c}$ are in A.P.

$\therefore \mathrm{b}^2=\mathrm{ac}$ ..... (i)

Also

$\begin{aligned} & \log _e\left(\frac{a}{2 b}\right), \log _e\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right) \text { are in A.P. } \\ & \left(\frac{2 b}{3 c}\right)^2=\frac{a}{2 b} \times \frac{3 c}{a} \\ & \frac{b}{c}=\frac{3}{2} \end{aligned}$

Putting in eq. (i) $b^2=a \times \frac{2 b}{3}$

$\begin{aligned} & \frac{\mathrm{a}}{\mathrm{b}}=\frac{3}{2} \\ & \mathrm{a}: \mathrm{b}: \mathrm{c}=9: 6: 4 \end{aligned}$

Let $r^{\prime}$th term of the GP be $a^{n-1}$. Given,

$\begin{aligned} & 2 a_r=a_{r+1}+a_{r+2} \\ & 2 a r^{n-1}=a r^n+a r^{n+1} \\ & \frac{2}{r}=1+r \\ & r^2+r-2=0 \end{aligned}$

Hence, we get, $r=-2$ (as $r \neq 1$)

So, $\mathrm{S}_{20}-\mathrm{S}_{18}=$ (Sum upto 20 terms) $-$ (Sum upto 18 terms) $=\mathrm{T}_{19}+\mathrm{T}_{20}$

$\mathrm{~T}_{19}+\mathrm{T}_{20}=\mathrm{ar}^{18}(1+\mathrm{r})$

Putting the values $\mathrm{a}=\frac{1}{8}$ and $\mathrm{r}=-2$;

we get $T_{19}+T_{20}=-2^{15}$

$\begin{aligned} & a+a r+a r^2+a r^3+\ldots .+a r^{63} \\ & =7\left(a+a r^2+a r^4 \ldots .+a r^{62}\right) \\ & \Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7 a\left(1-r^{64}\right)}{1-r^2} \\ & r=6 \end{aligned}$

$\begin{aligned} & a_6=2 \Rightarrow a+5 d=2 \\ & a_1 a_4 a_5=a(a+3 d)(a+4 d) \\ & =(2-5 d)(2-2 d)(2-d) \\ & f(d)=8-32 d+34 d^2-20 d+30 d^2-10 d^3 \\ & f^{\prime}(d)=-2(5 d-8)(3 d-2) \end{aligned}$

$\mathrm{d}=\frac{8}{5}$

$20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots \ldots,-129 \frac{1}{4}$

This is A.P. with common difference

$\begin{aligned} & d_1=-1+\frac{1}{4}=-\frac{3}{4} \\ & -129 \frac{1}{4}, \ldots \ldots \ldots \ldots . . .19 \frac{1}{4}, 20 \end{aligned}$

This is also A.P. $\mathrm{a}=-129 \frac{1}{4}$ and $\mathrm{d}=\frac{3}{4}$

Required term $=$

$\begin{aligned} & -129 \frac{1}{4}+(20-1)\left(\frac{3}{4}\right) \\ & =-129-\frac{1}{4}+15-\frac{3}{4}=-115 \end{aligned}$

$4,9,14,19, \ldots$, up to $25^{\text {th }}$ term

$\mathrm{T}_{25}=4+(25-1) 5=4+120=124$

$3,6,9,12, \ldots$, up to $37^{\text {th }}$ term

$\mathrm{T}_{37}=3+(37-1) 3=3+108=111$

Common difference of $\mathrm{I}^{\text {st }}$ series $\mathrm{d}_1=5$

Common difference of $\mathrm{II}^{\text {nd }}$ series $\mathrm{d}_2=3$

First common term $=9$, and their common difference $=15\left(\operatorname{LCM}\right.$ of $\mathrm{d}_1$ and $\left.\mathrm{d}_2\right)$ then common terms are $9,24,39,54,69,84,99$

Now, we have the following relations :

Arithmetic progression :

Since $A_1$ and $A_2$ are arithmetic means between $a$ and $b$, we can say that $a$, $A_1$, $A_2$, and $b$ are in an arithmetic progression. This means there are three equal intervals between $a$ and $b$, which are represented by the common difference $d$.

To find the value of $d$, we can use the following equation :

$ b - a = 3d $

From this equation, we can find the value of $d$ :

$ d = \frac{b - a}{3} $

$ A_1 = a + \frac{b - a}{3} = \frac{2a + b}{3} $

$ A_2 = \frac{a + 2b}{3} $

$ A_1 + A_2 = a + b $

Geometric progression :

$ a, G_1, G_2, G_3, b \text{ are in G.P. } $

$ r = \left(\frac{b}{a}\right)^{\frac{1}{4}} $

$ G_1 = \left(a^3b\right)^{\frac{1}{4}} $

$ G_2 = \left(a^2b^2\right)^{\frac{1}{4}} $

$ G_3 = \left(ab^3\right)^{\frac{1}{4}} $

We have the expression :

$ G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2 = a^3b + a^2b^2 + ab^3 + \left(a^3b\right)^{\frac{1}{2}}\cdot\left(ab^3\right)^{\frac{1}{2}} $

Simplify the expression :

$ a^3b + a^2b^2 + ab^3 + ab(a^2b^2) $

Factor out $ab$:

$ ab(a^2 + ab + b^2 + a^2b^2) $

Combine the terms :

$ ab(a^2 + 2ab + b^2) $

Rewrite the expression using the sum of squares :

$ ab(a + b)^2 $

Now, recall that $A_1 + A_2 = a + b$. Substitute this into the expression :

$ G_1 \cdot G_3 \cdot (A_1 + A_2)^2 $

Given the conditions :

1. $a_6 + a_8 = 2 \Rightarrow a r^5 + a r^7 = 2$
2. $a_3 \cdot a_5 = \frac{1}{9} \Rightarrow a^2 \cdot r^2 \cdot r^4 = \frac{1}{9} \Rightarrow a r^3 = \frac{1}{3}$

From this, we can form the equation $\frac{r^2}{3} + \frac{r^4}{3} = 2$, which simplifies to $r^4 + r^2 = 6$.

This can be factored to give $\left(r^2 - 2\right)\left(r^2 + 3\right) = 0$, yielding $r^2 = 2$ (since $r^2$ cannot be $-3$ for real $r$).

So, we have $r = \sqrt{2}$.

Substituting $r = \sqrt{2}$ into the equation $a r = \frac{1}{6}$, we get $a = \frac{1}{6\sqrt{2}}$.

Now, we find the value of $6(a_2+a_4)(a_4+a_6)$:

$6(a_2+a_4)(a_4+a_6) = 6\left(a r + a r^3\right)\left(a r^3 + a r^5\right)$

$= 6\left(\frac{1}{6\sqrt{2}} + \frac{1}{3\sqrt{2}}\right)\left(\frac{1}{3\sqrt{2}} + \frac{2}{3\sqrt{2}}\right)$

$= 6 \cdot \frac{1}{2} \cdot 1 = 3$.