iCON Education - Sequences and Series TS-EAMCET Test

Question 1: Palette 1/22

TS-EAMCET 2025 MCQ

$t_1, t_2, t_3, \ldots, t_n$ are positive integers, $S_n=t_1+t_2+t_3+\ldots+t_n$, $S_1=1^2, S_2=3^2, S_3=6^2, S_4=10^2, S_5=15^2$ and similarly other terms are there. Following this pattern, if $S_{10}=k^2$ then $k=$

Question 2: Palette 2/22

TS-EAMCET 2025 MCQ

$K=\left|\begin{array}{cc}3 & 4 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}1 & -1 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{3} & \frac{1}{4} \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{9} & -\frac{1}{16} \\ 5 & 4\end{array}\right|+\ldots$ to $\infty$, then $K=$

Question 3: Palette 3/22

TS-EAMCET 2025 MCQ

The value of the greatest integer $k$ satisfying the inequation $2^{n+4}+12 \geq k(n+4)$ for all $n \in N$ is

Question 4: Palette 4/22

TS-EAMCET 2025 MCQ

If $\frac{1}{2 \cdot 7}+\frac{1}{7 \cdot 12}+\frac{1}{12 \cdot 17}+\frac{1}{17 \cdot 22}+\ldots$ to 10 terms $=k$, then $k=$

$\frac{2}{51}$

$\frac{5}{51}$

$\frac{5}{52}$

$\frac{1}{26}$

Question 5: Palette 5/22

TS-EAMCET 2025 MCQ

The value of the greatest positive integer $k$, such that $49^k+1$ is a factor of $48\left(49^{125}+49^{124}+\ldots+49^2+49+1\right)$ is

Question 6: Palette 6/22

TS-EAMCET 2025 MCQ

$1+(1+3)+(1+3+5)+(1+3+5+7)+\ldots$ to 10 terms $=$

385

285

506

406

Question 7: Palette 7/22

TS-EAMCET 2024 MCQ

If $1 \cdot 3 \cdot 5+3 \cdot 5 \cdot 7+5 \cdot 7 \cdot 9 \ldots$ to $n$ terms $=n(n+1) f(n)$, then $f(2)=$

Question 8: Palette 8/22

TS-EAMCET 2024 MCQ

Assertion (A) : $1+\frac{2 \cdot 1}{3 \cdot 2}+\frac{2 \cdot 5}{3 \cdot 6} \frac{1}{4}+\frac{2 \cdot 5 \cdot 8}{3 \cdot 6 \cdot 9} \frac{1}{8}+\ldots \infty=\sqrt[3]{4}$

Reason (R) : |x| < 1,(1-x) $=1+n x+\frac{n(n+1)}{1 \cdot 2} x^2+\frac{n(n+1)(n+2)}{1 \cdot 2 \cdot 3} x^{3}+\ldots$

The correct answer is :

(A) and (R) are correct, ( $R$ ) is the correct explanation of $(A)$

(A) and (R) are correct, but (R) is not correct explanalion of (A)

(A) is correct but (R) is not correct

(A) is not correct but (R) is correct

Question 9: Palette 9/22

TS-EAMCET 2024 MCQ

Among the following four statements, the statement which is not true, for all $n \in N$ is

$(2 n+7)<(n+3)^2$

$1^2+2^2+\ldots \ldots+n^2>\frac{n^3}{3}$

$3 \cdot 5^{2 n+1}+2^{3 n+1}$ is divisible by 23

$2+7+12+\ldots \ldots+(5 n-3)=\frac{n(5 n-1)}{2}$

Question 10: Palette 10/22

TS-EAMCET 2024 MCQ

$\frac{1}{3 \cdot 6}+\frac{1}{6 \cdot 9}+\frac{1}{9 \cdot 12}+\ldots \ldots .$. to 9 terms $=$

$\frac{10}{99}$

$\frac{11}{108}$

$\frac{1}{10}$

$\frac{1}{90}$

Question 11: Palette 11/22

TS-EAMCET 2024 MCQ

When $|x|<2$, then coefficient of $x^2$ in the power series expansion of $\frac{x}{(x-2)(x-3)}$, is

$\frac{1}{6}$

$\frac{5}{36}$

$\frac{25}{216}$

$\frac{5}{18}$

Question 12: Palette 12/22

TS-EAMCET 2023 MCQ

If the roots of the equation $k x^3-18 x^2-36 x+8=0$ are in harmonic progression, then $k=$

Question 13: Palette 13/22

TS-EAMCET 2023 MCQ

If $f(x)$ is a function such that $f(x+y)=f(x)+f(y)$ and $f(1)=7$, then $\sum_{r=1}^n f(r)=$

$\frac{7 n}{2}$

$\frac{7(n+1)}{2}$

$7 n(n+1)$

$\frac{7 n(n+1)}{2}$

Question 14: Palette 14/22

TS-EAMCET 2023 MCQ

If $i=\sqrt{-1}$, then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

$\frac{9-3 i}{10}$

$9-3 i$

$9+3 i$

$\frac{9+3 i}{10}$

Question 15: Palette 15/22

TS-EAMCET 2023 MCQ

If $3 x=1+\frac{5}{8}+\frac{5}{8} \cdot \frac{9}{13}+\frac{5}{16}+\ldots$, then $x^4+4 x^3+6 x^2+4 x=$

Question 16: Palette 16/22

TS-EAMCET 2023 MCQ

The roots of the equation $x^3-14 x^2+56 x-64=0$ are in

arithmetic-geometric progression

harmonic progression

arithmetic progression

geometric progression

Question 17: Palette 17/22

TS-EAMCET 2022 MCQ

If $\alpha, \beta, \gamma$ are the roots of the equation $3 x^3-26 x^2+52 x-24=0$ such that $\alpha, \beta, \gamma$ are in geometric progression and $\alpha<\beta<\gamma$, then $3 \alpha+2 \beta+\gamma=$

$68 / 3$

$56 / 3$

Question 18: Palette 18/22

TS-EAMCET 2020 MCQ

Let $f(n)=A(-2)^n+B(-3)^n \forall A, B \in \mathbf{R}$ and $n \in \mathbf{N}-\{1,2\}$. If $f(n)+a f(n-1)+b f(n-2)=0$, then $(a+b)(b-a)=$

Question 19: Palette 19/22

TS-EAMCET 2020 MCQ

If $1+\frac{\cos \theta}{2}+\frac{\cos 2 \theta}{4}+\frac{\cos 3 \theta}{8}+\ldots \ldots=\frac{a-2 \cos \theta}{5+b \cos \theta}$ for some $a, b \in \mathbf{R}$, then $(a-b)^2=$

125

Question 20: Palette 20/22

TS-EAMCET 2020 MCQ

If $S_n$ is the sum of the first $n$ terms of the series $1^2+2 \times 2^2+3^2+2 \times 4^2+5^2+2 \times 6^2+\ldots \infty$, then, when $n$ is even $S_n=$

$\frac{n(n+1)}{2}$

$\frac{n^2(n+1)}{2}$

$\frac{n(n+1)^2}{2}$

$\frac{n^2(n+2)}{2}$

Question 21: Palette 21/22

TS-EAMCET 2020 MCQ

If the roots of the equation, $8 x^3+6 p x^2+3 q x-27=0$ are in a geometric progression, then $q^2+9 p^2+6 p q+q / p=$

-3

-10

Question 22: Palette 22/22

TS-EAMCET 2020 MCQ

Let the greatest common divisor of $m, n$ be 1 . If $\frac{1}{1 \cdot 7}+\frac{1}{7 \cdot 13}+\frac{1}{13 \cdot 19}+\ldots \ldots$. upto 20 terms $=\frac{m}{n}$, then $5 m+2 n=$

325

330

342

337