Sequences & Series Questions - JEE

For all $n \in \mathbb{N}$, $1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n!$ is equal to

The sum of \[(x + 2)^{n-1} + (x + 2)^{n-2}(x + 1) + (x + 2)^{n-3}(x + 1)^2 + \ldots + (x + 1)^{n-1}\] is equal to

If $2p + 3q + 4r = 15$, then the maximum value of $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$ is

Three numbers form an increasing GP. If the middle number is doubled, then the new numbers are in AP. The common ratio of the GP is

Let a1, a2, a3, ... be an AP, such that\[\frac{a_1 + a_2 + \ldots + a_p}{a_1 + a_2 + a_3 + \ldots + a_q} = \frac{p}{q}; \quad p \neq q\]Then $\frac{a_6}{a_{21}}$ is equal to:

The coefficient of $x^{n-2}$ in the polynomial $(x + 1)(x + 2)(x + 3)\cdots(x + n)$ is

Let $x,y,z$ be three natural numbers such that $x+y+z=10$. The maximum possible value of $xyz+xy+yz+zx$ is

Suppose x and y are two real numbers such that the rth mean between x and 2y is equal to rth mean between 2x and y, when n arithmetic means are inserted between them in both the cases. Then $\frac{n+1}{r} - \frac{y}{x}$ is equal to

Let $a$, $b$, $c$, $d$ and $e$ be positive real numbers such that $a + b + c + d + e = 15$ and $ab^2c^3d^4e^5 = (120)^3 \times 50$. Then the value of $a^2 + b^2 + c^2 + d^2 + e^2$ is ______.

The length of three unequal edges of a rectangular solid block are in GP. The volume of the block is 216 cm³ and the total surface area is 252 cm². The length of the longest edge is

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is __________.

Let a, b, c, d and p be any non-zero distinct real numbers such that $(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$. Then

Let n ∈ ℕ, n ≠ 25. If A, G and H denote the arithmetic mean, geometric mean and harmonic mean of 25 and n. Then, the least value of n for which A, G, H ∈ {25, 26, ..., n}, is

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Let $b_i > 1$ for $i = 1, 2, \ldots, 101$. Suppose $\log_e b_1, \log_e b_2, \ldots, \log_e b_{101}$ are in arithmetic progression (A.P.) with the common difference $\log_e 2$. Suppose $a_1, a_2, \ldots, a_{101}$ are in A.P. such that $a_1 = b_1$ and $a_{51} = b_{51}$. If $t = b_1 + b_2 + \cdots + b_{51}$ and $s = a_1 + a_2 + \ldots + a_{51}$, then

Divide 20 in four parts which are in AP such that the product of the first and the fourth is to the product of the second and the third = 2 : 3.

Let f be defined on the natural numbers as follows: f(1) = 1 and for n ≥ 1, f(n) = f[f(n − 1)] + f[n − f(n − 1)]. The value of $\sum_{r=1}^{30} f(r)$ is

The sum of the first 20 terms common between the series $3 + 7 + 11 + 15 + \ldots$ and $1 + 6 + 11 + 16 + \ldots$ is

1 1 1 K 22. then the remainder when K is divided by 6 is 2.3% | 22.39 “910.3 210,310” 12. Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is ——, [EE (Main) 2022] then the greatest number amongst them is: [JEE (Main) 2020] @1 (2)2 (3) 3 (4)5

The sum of the following series 9(1°+2?+3") 12(17+2°+3%+4*) 15(1° +2? +... +5?) 1+6+ 9 11 up to 15 terms, is: [EE (Main) 2019] (1) 7820 (2) 7830 (3) 7520 (4) 7510

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.If $a$, $b$, and $c$ are roots of the equation $x^3 + qx^2 + rx + s = 0$, then the value of $r$ is

If the arithmetic progression whose common difference is non-zero, the sum of first $3n$ terms is equal to the sum of the next $n$ terms. The ratio of the sum of the first $2n$ terms to the next $2n$ terms is

Let $P(n) = \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$. Consider the following statements:Statement-1: $P(n) = \sqrt{n(n+1)} Statement-2: \(P(k+1) > \sqrt{k+1}$ whenever $P(k) > \sqrt{k}$Which of the following is correct?

For Problems 25–27: Two arithmetic progressions have the same numbers. The ratio of the last term of the first progression to the first term of the second progression is equal to the ratio of the last term of the second progression to the first term of the first progression and is equal to 4. The ratio of the sum of the $n$ terms of the first progression to the sum of the $n$ terms of the second progression is equal to 2.The ratio of their common difference 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

Find the sum: \[1 + \dfrac{1 + \dfrac{1}{1!}}{2} + \dfrac{1 + \dfrac{1}{1!} + \dfrac{1}{2!}}{2^2} + \dfrac{1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!}}{2^3} + \ldots \text{ to } \infty.\]

897. Let $A_r$, $r = 1, 2, \ldots, 29$ be arithmetic means between 303 and $-57$ where $A_r > A_{r+1}$ $\forall$ $r = 1, 2, \ldots, 28$. If $S$ be the sum of these means, then the value of $\left[\dfrac{S}{(A_{14}-12)|A_r|_{\min}}\right]$.[Note: $[k]$ denotes greatest integer less than or equal to $k$ and $|A_r|_{\min}$ denotes the minimum value of $|A_r|$.]

If a, a1, a2, a3, ..., a2n, b are in AP and a, b1, b2, b3, ..., b2n, b are in GP and h is the HM of a and b, then$\frac{a_1 + a_{2n}}{b_1 b_{2n}} + \frac{a_2 + a_{2n-1}}{b_2 b_{2n-1}} + \ldots + \frac{a_n + a_{n+1}}{b_n b_{n+1}}$ is equal to

Let $E = \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots$. Then,

(4) (20°, 6) (2) (10%, 6) (3) (102, 3) (4) (10, 9) 10. If the sum of the first 40 terms of the series, 3+4+8+9+13+14+18+19 +... is (102)m, then m MSSO050 is equal to: [JEE (Main) 2020] 20. Let a, = b; = 1,a, = a,_1+ 2andb, =a, + b, _, for every natural number n> 2. Then Ya, :b, (1) 20 (2)5 (3) 10 (4) 25 n=l

If $\frac{1}{b-c}$, $\frac{1}{c-a}$, $\frac{1}{a-b}$ be consecutive terms of an AP, then $(b-c)^2$, $(c-a)^2$, $(a-b)^2$ will be in

Let three positive numbers $a, b, c$ (in order) be in HP such that $a + c = 8$. If $\langle t_n \rangle$ is a geometric progression with common ratio 3 where $t_1 = a - \dfrac{b}{2}$, $t_2 = \dfrac{b}{2}$ and $t_3 = c - \dfrac{b}{2}$, then find the value of $t_7\left(\dfrac{2}{3}\right)^6$.

Let {a,, }%=1 be a sequence such that a, = 1, az = 1 and dy42 = 2an41 + Gp for all n > 1. Then the 4, value of 47> 3 in is equal to —_—_— [EE (Main) 2021] 26. The 4% term of G.P. is 500 and its common ratio is —, me N . Let S,, denote the sum of the first n m n=

The 4% term of G.P. is 500 and its common ratio is i , méN .Let S, denote the sum of the first n n=1 m — f this GP. If Ss > Ss + 1 and S7

Let $\alpha, \beta$ are the roots of the quadratic equation $2x^2 - 5x + 1 = 0$. If $S_n = (\alpha)^{2n} + (\beta)^{2n}$, then find the value of $\dfrac{4S_{2021} + S_{2019}}{S_{2020}}$.

If \[ S = \sum_{r=1}^{\infty} \frac{r^3 + (r^2+1)^2}{(r^4+r^2+1)(r^2+r)} = \frac{a}{b} \] (in lowest terms), find $ a + b $.

Find the sum of the first 19 terms of the AP $a_1, a_2, a_3, \ldots$ if it is known that $a_1 + a_8 + a_{12} + a_{19} = 224$.

The sum of the series $1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \ldots$ upto 11th term is __________.

Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 − S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is 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 sum of all the four numbers is

Let $a_n = \dfrac{3^n}{n}$. Then the sum $S = \displaystyle\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \dfrac{1}{(a_m)(a_m + a_n)}$ equals ________.

The sum $3 + 8 + 16 + 27 + 41 + ... $ upto 20 terms is equal to

If $a$, $|a-1|$ and $|a-2|$ are the first three terms of an arithmetic progression, then its sum upto 20 terms is

Let $R_1$ and $R_2$, respectively, be the maximum ranges up and down an inclined plane and R be the maximum range on the horizontal plane. The $R_1$, R, $R_2$ are in

The given series is $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \cdots$. If the sum of the first 20 terms is $A$ and the sum of the first 40 terms is $B$, then $B - 2A = 100\lambda$. Find $\lambda$.

If a, b, c are positive, a + b + c = 1 and the minimum value of (1 + 1/a)(1 + 1/b)(1 + 1/c) is k, then k is

(b) If the sum of the first $2n$ terms of the A.P. $2, 5, 8, \ldots$ is equal to the sum of the first $n$ terms of the A.P. $57, 59, 61, \ldots$, then $n$ equals

The sum to infinity of the series $1 + \frac{1}{2} + \frac{1}{2 \cdot 4} + \frac{1}{2 \cdot 4 \cdot 8} + ...$is

If the roots of the equation $10x^3 - cx^2 - 54x - 27 = 0$ are in harmonic progression, then the value of $c$ must be equal to

The GM between √9 and √16, is

Sequences & Series Questions (847)