Sequences & Series Questions (847)

Let \(\{a_n\}\) \((n \geq 1)\) be a sequence such that \(a_1 = 1\), and \(3a_{n+1} - 3a_n = 1\) for all \(n \geq 1\). Then find the value of \(a_{2002}\).

Let $a_1,a_2,2,a_3,a_4$ be in an AGP with common ratio 2 and sum $\frac{49}{2}$. Then $a_4$ is equal to ______________

Let $\displaystyle\sum_{n=0}^{\infty} \dfrac{n^3((2n)!) + (2n-1)(n!)}{(n!)((2n)!)} = ae + \dfrac{b}{e} + c$, where $a, b, c \in \mathbb{Z}$ and $e = \displaystyle\sum_{n=0}^{\infty} \dfrac{1}{n!}$. Then $a^2 - b + c$ is equal to ______.

The sum of the common terms of the following three arithmetic progressions: $3, 7, 11, 15, \ldots, 399$; $2, 5, 8, 11, \ldots, 359$; and $2, 7, 12, 17, \ldots, 197$, is equal to ______.

Let $0<z<y<x$ with $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ in AP and $x,\sqrt{2}y,z$ in GP. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z)^2$ is equal to

If $f(x)=\dfrac{(\tan1°)x+\ln123}{x\ln1234-(\tan1°)}$, $x>0$, then $f(f(x))+f\!\left(f\!\left(\dfrac{4}{x}\right)\right)$ has minimum value

For two positive numbers $a, b$, if $a, b$ and $\dfrac{1}{18}$ are in a geometric progression, while $\dfrac{1}{a}$, $10$ and $\dfrac{1}{b}$ are in an arithmetic progression, then $16a + 12b$ is equal to ______.

Let $a, b, c > 1$, $a^3$, $b^3$ and $c^3$ be in A.P., and $\log_a b$, $\log_c a$ and $\log_b c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\dfrac{a+4b+c}{3}$ and the common difference is $\dfrac{a-8b+c}{10}$ is $-444$, then $abc$ is equal to

For three positive integers p, q, r, $x^{pq^2} = y^{qr} = z^{p^2r}$ and $r = pq + 1$ such that $3, 3\log_y x, 3\log_z y, 7\log_x z$ are in A.P. with common difference $\dfrac{1}{2}$. Then $r - p - q$ is equal to

What is the maximum sum of the series \[20 + 19\frac{1}{3} + 18\frac{2}{3} + \ldots?\]

The 2008th term of the sequence \(1, \underbrace{2,2,2}_{3}, \underbrace{3,3,3,3,3}_{6}, \underbrace{4,4,4,4,4,4,4,4,4,4}_{10}, \ldots\) where \(n\) occurs \(\dfrac{n(n+1)}{2}\) times in the sequence, equals \(k\), then find \(\dfrac{k}{5}\).

Apack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from 33. Let the positive integers be written in the form: the pack and the sum of the numbers on the remaining cards is 1224. If the smaller to the numbers 1 on the removed cards is k, then k - 20 = [JEE (Advanced) 2013] 2 3

If \( (1-r)(1-2x-4x^2-8x^3-16x^4-32x^5) = 1 - r^6 \), \( (r \neq 1) \), then a value of \( \dfrac{r}{x} \) is

Let $S=\dfrac{10^9}{5}+\dfrac{10^8}{5^2}+\cdots+\dfrac{2}{5^{108}}+\dfrac{1}{5^{108}}$. Then $(16S-(25)^{-54})$ is equal to

Let \(P(x) = \sum_{k=1}^{n} kx^k \equiv n(x-a_1)(x-a_2)\cdots(x-a_n)\). If \(\sum_{k=1}^{n} \dfrac{1}{(1-a_k)^2} = 13\), find the value of \(n\).

Let $a_1 = b_1 = 1$ and $a_n = a_{n-1} + (n-1)$, $b_n = b_{n-1} + a_{n-1}$, $\forall n \geq 2$. If $S = \displaystyle\sum_{n=1}^{10} \dfrac{b_n}{2^n}$ and $T = \displaystyle\sum_{n=1}^{8} \dfrac{n}{2^{n-1}}$, then $2^7(2S - T)$ is equal to ______.

In an A.P., the sixth term $a_6=2$. If $a_1a_4a_5$ is the greatest, then the common difference of the A.P. is equal to

The 4th term of GP is 500 and its common ratio is $\dfrac{1}{m}$, $m \in \mathbb{N}$. Let $S_n$ denote the sum of the first n terms of this GP. If $S_6 > S_5 + 1$ and $S_7 < S_6 + \dfrac{1}{2}$, then the number of possible values of m is ______.

Let $A_1, A_2, A_3$ be three A.P.s with the same common difference $d$ and having their first terms as $A, A+1, A+2$ respectively. Let $a, b, c$ be the $7^{\text{th}}, 9^{\text{th}}, 17^{\text{th}}$ terms of $A_1, A_2, A_3$ respectively such that $\begin{vmatrix} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1 \end{vmatrix} + 70 = 0$. If $a = 29$, then the sum of first 20 terms of an AP whose first term is $c - a - b$ and common difference is $\dfrac{d}{12}$, is equal to ______.

Let $f(x)=\sum_{k=1}^{10}kx^k$. If $2f(2)+f'(2)=2^n\cdot m+1$ with $m$ odd, then $n$ is equal to

Find the value of \(\displaystyle\sum_{1 \le i

Let three consecutive terms of a G.P. be \(\dfrac{a}{r}\), \(a\) and \(ar\). Given \(\dfrac{a}{r} \times a \times ar = a^3 = 512\). If 4 is added to the first and second terms, the resulting terms together with the third term form an A.P. Find the sum of the three consecutive terms of G.P.

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is ______.

Let $S_K=\frac{1+2+\cdots+K}{K}$ and $\displaystyle\sum_{j=1}^n S_j^2=\frac{An(Bn^2+Cn+D)}{?}$ with $A,B,C,D\in\mathbb{N}$, $A$ least. Then

Let $\{a_k\}$ and $\{b_k\}$, $k \in \mathbb{N}$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $a_1 = b_1 = 4$ and $r_1 < r_2$. Let $c_k = a_k + b_k$, $k \in \mathbb{N}$. If $c_2 = 5$ and $c_3 = \dfrac{13}{4}$, then $\displaystyle\sum_{k=1}^{\infty} c_k - (12a_6 + 8b_4)$ is equal to ______.

If $\dfrac{1^3 + 2^3 + 3^3 + \ldots \text{ upto n terms}}{1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots \text{ upto n terms}} = \dfrac{9}{5}$, then the value of n is ______.

Let S be the sum of the first 9 terms of the series {x + ka} + {x2 + (k + 2)a} + {x3 + (k + 4)a} + {x4 + (k + 6)a} + ... where a ≠ 0 and x ≠ 1.If \(S = \frac{x^{10} - x + 45a(x-1)}{x-1}\), then k is equal to

If $\log_e a,\log_e b,\log_e c$ are in an A.P. and $\log_e a-\log_e 2b,\log_e 2b-\log_e 3c,\log_e 3c-\log_e a$ are also in an A.P., then $a:b:c$ is equal to

Let $a_1, a_2, \ldots, a_n$ be in A.P. If $a_5 = 2a_7$ and $a_{11} = 18$, then $12\left(\dfrac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \dfrac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \dfrac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$ is equal to ______.

If $\gcd(m,n)=1$ and $1^2-2^2+3^2-4^2+\cdots+(2023)^2=1012m^2n$, then $m^2-n^2$ is equal to

The sum $1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$ is ______.

The ratio of the sums of \(n\) terms of two APs is \((3n-13):(5n+21)\). Find the ratio of the 24th terms of the two progressions.

Let $a_1, a_2, a_3, \ldots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to ______.

If the set $R=\{(a,b):a+5b=42,\,a,b\in\mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^{m}(1-i^{n!})=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

Let $A_1$ and $A_2$ be two AMs and $G_1,G_2,G_3$ be three GMs of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to

Let $a_1 = 8$, $a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is ______.

Consider a cube, if all its six faces are assigned with a unique number from 2, 3, 4, 5, 6, 7 with one number at each face. For each of the eight vertices of the cube, a product of three numbers where the three numbers are the numbers assigned to the three faces that include the vertex. What is the largest value of the sum of these eight products?

36. If \((1-p)(1+3x+9x^2+27x^3+81x^4+243x^5) = 1 - p^6\), \(p \neq 1\), then the value of \(\dfrac{p}{x}\) is

If $(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\cdots+20(21)^{19}=k(20)^{19}$, then $k$ is equal to _____.

281. Given \( S_A = 2m + (2m+1) + (2m+2) + \cdots + 4m \) and \( S_B = (2m+1) + (2m+3) + (2m+5) + \cdots + (4m-1) \). If \( \dfrac{S_A}{S_B} = k + \dfrac{1}{l} \), find the value of \( k + l \).

Which of the following statements is a tautology?

Sum of all terms of AP $3,8,13,\ldots,373$ that are not divisible by 3 is

Let $x_1,x_2,\ldots,x_{100}$ be in AP, $x_1=2$, mean $=200$. If $y_i=i(x_i-i)$, then mean of $y_1,\ldots,y_{100}$ is

The parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then

The sum of the first 20 terms of the series $5+11+19+29+41+\ldots$ is

If the sum $\left(1-\dfrac{1}{2}\right)+\left(1-\dfrac{1}{2^2\cdot3}+\cdots\right)+\cdots=\dfrac{\alpha}{\beta}$, $\gcd(\alpha,\beta)=1$, then $\alpha+3\beta$ is equal to

A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?

Let $a,b,c,d>0$ with $a+b+c+d=11$. If max of $a^5b^3c^2d=3750\beta$, then $\beta$ is

312. Let \(S_n\) denote the sum of the first \(n\) terms of an AP. If \(S_{2n} = 3S_n\), then the ratio \(S_{3n} : S_n\) is equal to

Let $\langle a_{n}\rangle$ be a sequence such that $a_{0}=0$, $a_{1}=\dfrac{1}{2}$ and $2a_{n+2}=5a_{n+1}-3a_{n},\ n=0,1,2,\dots$ Then $\displaystyle\sum_{k=1}^{100}a_{k}$ is equal to: