Sequences & Series Questions - JEE

If the first term of an A.P. is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first $20$ terms is equal to:

54. Let $\alpha_n$, $\beta_n$ be the distinct roots of the equation $x^2 + (n+1)x + n^2 = 0$. If $\displaystyle\sum_{n=2}^{2021} \frac{1}{(\alpha_n + 1)(\beta_n + 1)}$ can be expressed in the form $\dfrac{a}{b}$, where $a$ and $b$ are positive integers, the value of $(b - a)$ is:

$\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\dfrac{k^3+6k^2+11k+5}{(k+3)!}$ equals

The interior angles of a polygon with $n$ sides are in an A.P.\ with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then $n$ is equal to \rule{2cm}{0.4pt}.

Let $S_{n}=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dots$ up to $n$ terms. If the sum of the first six terms of an A.P.\ with first term $-p$ and common difference $p$ is $\sqrt{2026\cdot S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P.\ is:

If $\displaystyle\sum_{r=1}^n T_r = \dfrac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\displaystyle\lim_{n\to\infty}\sum_{r=1}^n\dfrac{1}{T_r}$ equals

The value of $\displaystyle\sum_{r=1}^{\infty}\frac{8r}{4r^4+1}$ is

The sixth term of an AP is equal to 2. The value of the common difference of the AP which makes the product $a_1 a_4 a_5$ least, is given by

ABCD is a square of length a, $a \in \mathbb{N}$, $a > 1$. Let $L_1, L_2, L_3, \ldots$ be points on BC such that $BL_1 = L_1L_2 = L_2L_3 = \cdots = 1$ and $M_1, M_2, M_3, \ldots$ be points on CD such that $CM_1 = M_1M_2 = M_2M_3 = \cdots = 1$. Then $\sum_{n=1}^{a-1}(AL_n^2 + L_nM_n^2)$ is equal to

Let an increasing GP have first term $a_1$ and common ratio $r$. If $a_1 a_5 = 28$ and $a_2+a_4=29$, then $a_6$ equals

The sum of the infinite series $\frac{1}{9} + \frac{1}{18} + \frac{1}{30} + \frac{1}{45} + \frac{1}{63} + \cdots$

If the sum of the first 10 terms of the series 4.1$4 + 4.2$$4 + 4.3$4 +$\ldots is m n, where gcd$(m, n)$= 1, then$1+4.1$$1+4.2$$1+4.3$$$$m + n$is equal to______$

If the sum of the first 20 terms of the series 4.1 4.2 4.3 4.4 + + + +$\ldots 2 4 2 4 2 4 2 4$4+3.1 +1$$4+3.2 +2$$4+3.3 +3$$$$4+3.4 +4$is m n, where m and n are coprime, then$m + n$is equal to$:-

The sum$1 + 1+3$$1+3+5$$1+3+5+7$2! + 3! + 4! +$\ldots upto$$\infty terms, is equal to$

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by 21 2. Then the number of terms which are integers in the A.P. is:

Let x, x, x, x be in a geometric progression. 2, 7, 9, 5 are subtracted respectively from x, x, x x then the 1 2 3 4 1 2 3 4 resulting numbers are in an arithmetic progression. Then the value of 1 24$($$x_{1}$$$x_{2}$$$x_{3}$$$$$$x_{4}$$)$is:

Let $a_1, a_2, a_3, \ldots$ be a G. P. of increasing positive numbers. If $a_2 a_3 = 729$ and $a_3 + a_5 = \frac{111}{4}$, then $24(a_1+a_2+a_3)$ is equal to

If $a_1, a_2, \ldots, a_{4001}$ are in arithmetic progression and $\dfrac{1}{a_1 a_2} + \dfrac{1}{a_2 a_3} + \ldots + \dfrac{1}{a_{4000} a_{4001}} = 10$ and $a_2 + a_{4000} = 50$. Find the value of $|a_1 - a_{4001}|$.

If $\displaystyle\sum_{r=1}^{n}T_{r}=\dfrac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\dfrac{1}{T_{r}}$ is equal to:

For positive integers $n$, if $4a_{n}=(n^{2}+5n+6)$ and $S_{n}=\displaystyle\sum_{k=1}^{n}\dfrac{1}{a_{k}}$, then the value of $507\,S_{2025}$ is:

2 [IIT JEE 2009] 28. a= hn then a, + a, + + dzs is equal to: [EE (Main) 2023] 4n’ -16n+15’ (A) n(4n? —1)c? (8) n(4n? +1)c? (©) n(4n? -1)c? @) n(4n? +1)c?

If the sum $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots$ up to 20 terms is equal to $\frac{n}{21}$, then $n$ is equal to

For Problems 19–21: Let $A_1, A_2, A_3, \ldots, A_m$ be the arithmetic means between $-2$ and 1027 and $G_1, G_2, G_3, \ldots, G_n$ be the geometric means between 1 and 1024. The product of geometric means is $2^{45}$ and sum of arithmetic means is $1025 \times 171$.The numbers $2A_{171},\ G_5^2 + 1,\ 2A_{172}$ are in

23. Let $\{t_n\}$ be a sequence of integers in G.P. in which $t_4 : t_6 = 1:4$ and $t_2 + t_5 = 216$. Then $t_1$ is

If $1$, $a$, $9$ and $4$ are in harmonic progression, then the value of $a + b$ is equal to

If $q_1, q_2, q_3, \ldots, q_n$ are in AP, whose common difference is d, then $\sin d (\sec q_1 \sec q_2 + \sec q_2 \sec q_3 + \ldots + \sec q_{n-1} \sec q_n)$ is equal to

If a, B and y are three consecutive terms of a non-constant G.P. such that the equations ax? + 2Bx +y=0 and x? + x-1= 0 havea common root, then af + y) is equal to : [EE (Main) 2019] () By (2) 0 (3) ay (4) of

If $a_1, a_2, \ldots, a_n$ are in A.P. with common difference $d \neq 0$, then the sum of the series $d[\sec a_1 \sec a_2 + \sec a_2 \sec a_3 + \cdots + \sec a_{n-1} \sec a_n]$ is

Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\cdots$ up to 10 terms and $\beta=\displaystyle\sum_{n=1}^{10}n^4$. If $4\alpha-\beta=55k+40$, then $k$ is equal to

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to:

Let $4a_n = n^2+5n+6$ and $S_n = \displaystyle\sum_{k=1}^n\dfrac{1}{a_k}$. Then $507\,S_{2025}$ equals

Let $a_1, a_2, \ldots, a_{2024}$ be an AP. If $a_1+(a_5+a_{10}+a_{15}+\cdots+a_{2020})+a_{2024}=2233$, then $\displaystyle\sum_{i=1}^{2024}a_i$ equals

If $a$, $b$, $c$ are in AP and $(a+2b-c)(2b+c-a)(c+a-b)=\lambda abc$, then the value of $\lambda$ is

If $a, b, c, d \in R^+$ and $a, b, c, d$ are in H.P., then

Let the first term and common difference of an AP of positive integers be such that the sum of the first 3 terms is 54 and $1600 < S_{20} < 1800$. Then the $11^{\text{th}}$ term equals

The sum $\sum_{k=1}^{n} \frac{1}{(k+1)\sqrt{k} + k\sqrt{k+1}}$ is equal to:

The interior angles of a polygon are in AP with common difference $6°$. If the largest interior angle is $219°$, then the number of sides $n$ of the polygon is

The sum of the series $\left(1 \frac{1}{2}\right)^3 + \left(2 \frac{1}{2}\right)^3 + 3^3 + \left(3 \frac{1}{2}\right)^3 + \cdots$ to 10 terms is

The number of three-term increasing geometrical progressions comprising distinct natural numbers less than or equal to 100, with common ratio as a natural number, is

The sum $\sum_{k=1}^{n} \frac{k^2 - \frac{1}{2}}{k^4 + \frac{1}{4}}$ is equal to:

165. The sum of the infinite series $\dfrac{1}{9} + \dfrac{1}{18} + \dfrac{1}{30} + \dfrac{1}{45} + \dfrac{1}{63} + \ldots\ldots$

165. The sum of the infinite series $\dfrac{1}{9} + \dfrac{1}{18} + \dfrac{1}{30} + \dfrac{1}{45} + \dfrac{1}{63} + \ldots$ is:

If the AM of two positive numbers a and b (a ≠ b) is twice of their GM, then a : b is

If $b = ar$, $c = ar^2$, and $d = ar^3$, then $(b - c)^2 + (c - a)^2 + (d - b)^2$ is equal to

The number of $3$-digit numbers, that are divisible by $2$ and $3$, but not divisible by $4$ and $9$, is \rule{2cm}{0.4pt}.

Let $3,7,11,15,\ldots,403$ and $2,5,8,11,\ldots,404$ be two arithmetic progressions. Then the sum of the common terms in them, is equal to

Three numbers $a,\ ar,\ ar^2$ form a G.P. If $2(ar) = a + ar^2$ (i.e., the middle term is the AM of the other two), find $r$ given that the G.P. is increasing.

Let $\alpha,\beta$ be roots of $x^2-10x+2=0$. Value of $\dfrac{\alpha^{2028}+\beta^{2028}+8\alpha^{2022}+8\beta^{2022}}{\alpha^{2025}+\beta^{2025}}$ is

If sum of the series $1+\dfrac{\sqrt5-\sqrt3}{2\sqrt5}+\dfrac{8-2\sqrt{15}}{30}+\dfrac{14\sqrt5-18\sqrt3}{60\sqrt5}+\cdots=2+\dfrac{a+\sqrt{15}}{b}\log_b\!\left(\dfrac{a}{c}\right)$; $a,b,c\in\mathbb{N}$, $\gcd(a,b,c)=1$, then $2(a^2+b^2+c^2)$ is

Let $a_1, a_2, a_3, \ldots$ be in harmonic progression with $a_1 = 5$ and $a_{20} = 25$. If $n$ is the least positive integer for which $a_n < 0$, then the value of $4n - 100$ is

Sequences & Series Questions (847)