Sequences & Series Questions - JEE

$1 + 3 + 5 + 7 + 9$+$\ldots upto 40 terms is equal to 2 2$

If each term of a geometric progression $a_1,a_2,a_3,\ldots$ with $a_1=\dfrac{1}{8}$ and $a_2\neq a_1$, is the arithmetic mean of the next two terms and $S_n=a_1+a_2+\cdots+a_n$, then $S_{20}-S_{18}$ is equal to

MSSO55 14. The number of terms common to the two A-P.'s 3, 7, 11, w+ 407 and 2, 9, 16, , 709 is 2 3 24. The sum to 10 terms of the series eed 14242" 143° 43" we iste [EE (Main) 2023] [JEE (Main) 2020]

If p and q are positive real numbers such that $p^2 + q^2 = 1$, then the maximum value of $(p + q)$ is

The sum of the series $\dfrac{1}{1-3\cdot1^2+1^4}+\dfrac{2}{1-3\cdot2^2+2^4}+\dfrac{3}{1-3\cdot3^2+3^4}+\cdots$ up to 10 terms is

Let $3,a,b,c$ be in A.P. and $3,a-1,b+1,c+9$ be in G.P. Then the arithmetic mean of $a,b$ and $c$ is:

The interior angles of a polygon are in AP. If the smallest angle be 120° and the common difference be 5, then the number of sides is

Minimum possible area of the triangle is:

Maximum possible perimeter of the triangle is:

Find the number of common terms in the following sequences: (i) 3, 7, 11, … to 100 terms and 2, 5, 8, … to 100 terms.

If $S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}$, $x\neq0$, and $(60)^2S(60)=a(b)^b+b$, where $a,b\in\mathbb{N}$, then $(a+b)$ equal to

For Problems 10–12: Four different integers form an increasing A.P. One of these numbers is equal to the sum of the squares of the other three numbers.The product of all numbers is

If the roots of the equation $x^3 - 12x^2 + 39x - 28 = 0$ are in AP, then their common difference will be

Let $S_n$ denote the sum of first $n$ terms of the arithmetic sequence $\{a_n\}$. If $S_6 > S_7 > S_5$, then the value of integral value of $n$ which satisfy \(S_n S_{n+1}

14+34+5+ upto n terms 20 and 4+7+10+...upto n terms 7 7logy) x 1 1 1 n= log, x+log,, x? +log,, x* +log,, x® +... +00, then x is equal to

It is given that the numbers $3, 3\log_y x, 3\log_z y, 7\log_x z$ form an arithmetic progression. Then

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 1 + 2.2^2 + 3^2 + 2.4^2 + 5^2 + 2.6^2 + \ldotsIf B - 2A = 100\lambda, then \lambda is equal to

(a) Consider an infinite geometric series with first term 'a' and common ratio r. If the sum is 4 and the second term is 3/4, then:

The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots$ is:

If 1, $\log_9(3^{1-x} + 2)$, $\log_3(4 \cdot 3^x - 1)$ are in A.P., then $x$ equals

Fifth term of a G.P. is 2, then the product of its 9 terms is

The sum $\frac{1 \cdot 9}{1 \cdot 2 \cdot 3} + \frac{1 \cdot 28}{2 \cdot 3 \cdot 4} + \frac{1 \cdot 39}{3 \cdot 4 \cdot 5} + \frac{1 \cdot 522}{4 \cdot 5 \cdot 6} + \ldots$ up to infinite terms is equal to

If a, b, c are in H.P., then [expression] is equal to

Let p, q be integers and let α, β be the roots of the equation x2 − x − 1 = 0, where α ≠ β. For n = 0, 1, 2, ..., let an = pαn + qβn. If a4 = 28, then p + 2q =

(c) Let the positive numbers $a, b, c, d$ be in A.P. Then $abc, abd, acd, bcd$ are

Let $T_r$ be the $r$th term of an AP whose first term is $a$ and common difference is $d$. If for some positive integers $m, n, m \neq n$, $T_m = \dfrac{1}{n}$ and $T_n = \dfrac{1}{m}$, then $a - d$ equals

If $\sum_{r=1}^{n} r^4 = I(n)$, then $\sum_{r=1}^{n} (2r-1)^4$ is equal to

Let l, G1, G2, G3, n are in GP where $m = \dfrac{l+n}{2}$; $(l, n > 1)$. Then $(G_1)^4 + 2(G_2)^4 + (G_3)^4$ equals:

If $10(10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + \ldots + 10(11)^9 = k(10)^9$, then $k$ is equal to

The value of $\dfrac{1}{2!} + \dfrac{1}{4!} + \dfrac{1}{6!} + \cdots$ is:

The product of n positive numbers is unity. Their sum is

The sum of the 3rd and 4th terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7th term is

Consider an A.P. $a_1, a_2, a_3, \ldots$ such that $a_3 + a_5 + a_8 = 11$ and $a_4 + a_2 = -2$, then the value of $a_1 + a_6 + a_7$ is

For Problems 28–30: The numbers $a$, $b$, and $c$ are between 2 and 18, such that (i) their sum is 25, (ii) the numbers 2, $a$, and $b$ are consecutive terms of an A.P., (iii) the numbers $b$, $c$, 18 are consecutive terms of a G.P.The value of $abc$ is

If $\displaystyle\sum_{n=1}^{5} \dfrac{1}{n(n+1)(n+2)(n+3)} = \dfrac{k}{3}$, then $k$ is equal to

If $a, b, c$ are in HP, then the straight line $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{1}{c} = 0$ always passes through a fixed point. That point is:

The 1st, 2nd and 3rd terms of an arithmetic series are $a$, $b$ and $a^2$, where $a$ is negative. Then the sum of an infinite geometric series whose first three terms are $a$, $a^2$ and $b$ respectively, is:

If $\log_2(5 \times 2^x + 1)$, $\log_4(2^{1-x} + 1)$ and 1 are in A.P., then x equals

Let $S_n = \dfrac{1}{1^3} + \dfrac{1+2}{1^3+2^3} + \dfrac{1+2+3}{1^3+2^3+3^3} + \ldots\ldots (n \text{ terms})$, where $n = 1,2,3,4,\ldots\ldots$, then $S_n$ is always less than:

Let $a_1,\dfrac{a_2}{2},\dfrac{a_3}{2^2},\ldots,\dfrac{a_{10}}{2^9}$ be a G.P. of common ratio $\dfrac{1}{\sqrt{2}}$. If $a_1+a_2+\ldots+a_{10}=62$, then $a_1$ is equal to:

For $a, b > 0$, let $5a - b$, $2a + b$, $a + 2b$ be in A.P. and $(b+1)^2$, $ab+1$, $(a-1)^2$ are in G.P., then the value of $(a^{-1} + b^{-1})$ is ___.

In a non constant arithmetic progression having odd number of terms, having positive integral common difference, the ratio of the sum of the 1st, 3rd, 5th, 7th, ........... terms to the sum of remaining terms is 13 : 12, then the number of terms in the arithmetic progression, is:

If three positive numbers a, b and c are in AP such that abc = 8, then the minimum possible value of b is

If $H_n = 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n}$, then the value of $S_n = 1 + \dfrac{3}{2} + \dfrac{5}{3} + \cdots + \dfrac{99}{50}$ is

Let $a$, $b$, $c$, $d$ be four distinct real numbers in A.P. Then the smallest positive value of $k$ satisfying $2(a-b) + k(b-c)^2 + (c-a)^3 = 2(a-d) + (b-d)^2 + (c-d)^3$ is ___.

If S1, S2, S3, …, Sn are the sums of infinite geometric series whose first terms are 1, 2, 3, …, n and whose common ratios are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{n+1}$ respectively, then find the value of $\displaystyle\sum_{r=1}^{2n-1} S_r^2$.

If $a, b, c, d$ are distinct integers in AP such that $d = a^2 + b^2 + c^2$, then find $\dfrac{(a+b+c+d)}{5}$.

If $a_1, a_2, \ldots, a_n$ are in H.P., then $\dfrac{a_1}{a_2 + a_3 + \cdots + a_n},\ \dfrac{a_2}{a_1 + a_3 + \cdots + a_n},\ \ldots,\ \dfrac{a_n}{a_1 + a_2 + \cdots + a_{n-1}}$ are in

Consider an A.P.: $a_1,a_2,\ldots,a_n$; $a_1>0$. If $a_2-a_1=-\dfrac{3}{4}$, $a_n=\dfrac{1}{4}a_1$, and $\displaystyle\sum_{i=1}^n a_i=\dfrac{525}{2}$, then $\displaystyle\sum_{i=1}^{17}a_i$ is equal to

Find the sum $11^2 - 1^2 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2$.

Sequences & Series Questions (847)