Sequences & Series Questions - JEE

If A1, A2; G1, G2 and H1, H2 are two arithmetic, geometric and harmonic means, respectively between two quantities a and b, then which of the following is not the value of ab?

35. If $(1+x)(1+x^2)(1+x^4)\cdots(1+x^{128}) = \displaystyle\sum_{r=0}^{n} x^r$, then $n$ is equal to

If x, 2y and 3z are in AP, where the distinct numbers x, y, z are in GP, then the common ratio of the GP is

33. If $a, b$ and $c$ are in G.P. and $x, y$, respectively, are the arithmetic means between $a, b$ and $b, c$, then the value of $\dfrac{a}{x} + \dfrac{c}{y}$ is

If $b$ is the first term of an infinite G.P. whose sum is five, then $b$ lies in the interval

Let three real numbers $a,b,c$ be in arithmetic progression and $a+1,b,c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a,b$ and $c$ is 8, then the cube of the geometric mean of $a,b$ and $c$ is

If $1, \frac{1}{a}, \frac{1}{b}, \ldots, \frac{1}{c}$ are in A.P., find $a_{41}b_{10}$.

Let the A.P. be $a, a+d, a+2d, \ldots$ Given, $(a+d)(a+8d) = (a+4d)^2$

$\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n = 1 + \frac{1}{3} + \frac{1}{9} + \cdots \infty$

How many terms of $1 + 3 + 5 + 7 + \ldots$ amount to 1234321?

Let $a_1, a_2, \ldots, a_{30}$ be an A.P., $S = \displaystyle\sum_{i=1}^{30} a_i$ and $T = \displaystyle\sum_{i=1}^{15} a_{(2i-1)}$. If $a_5 = 27$ and $S - 2T = 75$, then $a_{10}$ is equal to __________.

317. $\frac{a^n + b^n}{a^{n-1} + b^{n-1}}$ is the AM between $a$ and $b$ if $n$ is

318. If $H$ be the harmonic mean between $P$ and $Q$, then the value of $\frac{H}{P} + \frac{H}{Q}$ is

Find the sum of all 3-digit natural numbers which are of the form $3m + 2$, $m \in \mathbb{N}$, i.e., leaves the remainder 2 when divided by 3.

Find the sum of all numbers from 100 to 550 which are divisible by 9.

The sum of the following series $1 + 6 + \dfrac{9(1^2+2^2+3^2)}{7} + \dfrac{12(1^2+2^2+3^2+4^2)}{9} + \dfrac{15(1^2+2^2+\cdots+5^2)}{11} + \cdots$ upto 15 terms, is __________.

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then:

40. Let $a_n$ be the $n$th term of a G.P. of positive numbers. Let $\displaystyle\sum_{n=1}^{100} a_{2n} = \alpha$ and $\displaystyle\sum_{n=1}^{100} a_{2n-1} = \beta$, such that $\alpha \neq \beta$, then the common ratio is

For an AP of odd number of terms, the sum of all the terms is $\dfrac{15}{8}$ times the sum of the terms in odd places. Find the number of terms in the AP.

Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\dfrac{a}{c}$ is equal to __________.

The equation $\displaystyle\sum_{r=1}^{\infty} \dfrac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)} = \dfrac{a}{b}$ holds true for co-prime positive integers $a$ and $b$. Find $a+b$.

32. If the $p$th, $q$th, $r$th, and $s$th terms of an A.P. are in G.P., then $p - q, q - r, r - s$ are in

Find four numbers between 4 and 40 so that the six numbers are consecutive terms of an AP.

313. Sum of the first $n$ terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \ldots$ is equal to

Find the number of triplets (a, b, c) such that a, b, c are three distinct positive numbers and a, b, c, b + c – a, c + a – b, a + b – c and a + b + c form a seven term arithmetic progression in some order.

The 8th common term of the series $S_1 = 3 + 7 + 11 + 15 + 19 + \ldots$ and $S_2 = 1 + 6 + 11 + 16 + 21 + \ldots$ is ______.

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3,7,11,\ldots$ If $40<\dfrac{6}{n(n+1)}\displaystyle\sum_{k=1}^{n}S_k<42$, then $n$ equals

Statement 1: $\sum_{k=1}^{\infty}\dfrac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})}=2$. Statement 2: $\sum_{k=1}^{n}[k^3-(k-1)^3]=n^3$ for any natural number $n$.

27. If $x, y, z$ are in G.P. and $a^x = b^y = c^z$, then

316. If the first and $(2n-1)$th terms of an AP, a GP and an HP are equal and their $n$th terms are $a$, $b$ and $c$ respectively, then

166. If $a+c,\ a+b,\ b+c$ are in G.P. and $a, c, b$ are in H.P. where $a, b, c > 0$, then the value of $\dfrac{a+b}{c}$ is:

22. If $a, b, c, d$ are in G.P., then $(b-c)^2 + (c-a)^2 + (d-b)^2$ in equal to

$[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\cdots+[\sqrt{120}]$ is equal to

Let $\langle a_n\rangle$ with $\sum_{k=1}^n a_k=\frac{n^2+3n}{(n+1)(n+2)}$. If $28\sum_{k=1}^{10}\frac{1}{a_k}=p_1p_2\cdots p_m$, then $m$ is equal to

$2\cdot2^2-3^2+2\cdot4^2-5^2+2\cdot6^2-\ldots$ (20 terms) is equal to

Let $a_1,a_2,a_3,\ldots$ be an increasing GP. If $a_6+a_8=2$ and $a_3\cdot a_5=\frac{1}{9}$, then $6(a_2+a_4)(a_4+a_6)$ is equal to

Let the positive integers be written in the form: Row 1: 1; Row 2: 2,3; Row 3: 4,5,6; Row 4: 7,8,9,10; $\ldots$ If the $k$th row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is

28. The number of terms common between the series $1 + 2 + 4 + 8 + \cdots$ to 100 terms and $1 + 4 + 7 + 10 + \cdots$ to 100 terms is

Let a, a, a$\ldots be i_n a_n A.P. such that$$\sum 1 2 3 12$k=1$$a2k-1$= - 72 5$a_{1}$, a_{1}$$$\ne 0. If$$$\sum n$k=1$$ak = 0$, then n is$:

Let a be the n n th term of an A. P. If S$n =$$a_{1}$$+$$a_{2}$$+ a_{3}$+$\$ldots + an = 700$,$$a_{6}$= 7$and$S = 7$, then a is equal to$: 7 n

If $8=3+\dfrac{1}{4}(3+p)+\dfrac{1}{4^2}(3+2p)+\dfrac{1}{4^3}(3+3p)+\cdots\infty$, then the value of $p$ is

Suppose that the number of terms in an A.P.\ is $2k$, $k\in\mathbb{N}$. If the sum of all odd terms of the A.P.\ is $40$, the sum of all even terms is $55$, and the last term of the A.P.\ exceeds the first term by $27$, then $k$ is equal to:

The roots of the quadratic equation $3x^{2}-px+q=0$ are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with common difference $\dfrac{3}{2}$. If the sum of the first $11$ terms of this arithmetic progression is $88$, then $q-2p$ is equal to \rule{2cm}{0.4pt}.

In an arithmetic progression, if $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ is equal to:

Let $a$ and $b$ be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, be equal to $p^{\text{th}}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

The number of 3-digit natural numbers that are divisible by both 2 and 3 but not by both 4 and 9 is

Let $S_n$ denote the sum of the first $n$ terms of an AP. If $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ equals

The roots of the equation $3x^2-px+q=0$ are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an AP with common difference $\dfrac{3}{2}$ and $S_{11}=88$. Then $q-2p$ equals

Let the first term of an AP be 3. If the sum of its first 4 terms is $\dfrac{1}{5}$ of the sum of the next 4 terms, then $S_{20}$ equals

Let $a_0=0$, $a_1=\dfrac{1}{2}$, and $2a_{n+2}=5a_{n+1}-3a_n$ for $n\geq0$. Then $\displaystyle\sum_{k=1}^{100}a_k$ equals

Sequences & Series Questions (847)