Sum of Series: JEE Questions & Solutions

Let a1=1 and for n 1, an+1 = 12an + n^2-2n-12n^2(n+1)^2. Then n=1^ (an - 2n^2) is equal to:

The positive integer n, for which the solutions of the equation x(x+2) + (x+2)(x+4) + + (x+2n-2)(x+2n) = 8n3 are two consecutive even…

Let 729, 81, 9, 1, be a sequence and Pn denote the product of the first n terms of this sequence. If 2 n=1^40 (Pn)^ 1n = 3^ - 13^ and (,) =…

Let S= 12!5!+ 13!2!3!+ 15!2!1!+ up to 13 terms. If 13S= 2^kn!, k N, then n+k is equal to

The value of ( 13+ 47) + ( 13^2+ 13 47+ 47^2) + ( 13^3+ 13^2 47+ 13 47^2+ 47^3) + up to infinite terms is

The value of k=1^ (-1)^k+1 ( k(k+1)k!) is:

Evaluate: 63^26+ 10 13^25+ 10 23^24+ 10 2^23^23+ + 10 2^243.

If r=1^25 ( rr^4+r^2+1)= pq, where p and q are positive integers such that (p,q)=1, then p+q is equal to.

If r=1^n Tr = (2n-1)(2n+1)(2n+3)(2n+5)64, then n r=1^n 1Tr is equal to:

If 7 = 5 + 17(5 +) + 17^2(5 + 2) + 17^3(5 + 3) +, then the value of is:

Let an be a sequence such that a0 = 0, a1 = 12, and 2an+2 = 5an+1 - 3an. n= 0,1,2,3.... Then k=1^100 ak is equal to:

Let E1: x^29 + y^24 = 1 be an ellipse. Ellipses Ei are constructed such that their centers and eccentricities are the same as that of E1,…

For positive integers n, if 4 an = n^2 + 5n + 64 and Sn = k=1^n ( 1ak), then the value of 507 S2025 is:

The value of n ( k=1^n k^3 + 6k^2 + 11k + 5(k + 3)!) is:

Let S = N 0. Define a relation R from S to R by: R = (x, y): e y = x e ( 25), x S, y R. Then, the sum of all the elements in the range of R…

Let Pn = ^n + ^n, n N. If P10 = 123, P9 = 76, P8 = 47 and P1 = 1, then the quadratic equation having roots and 1 is:

If the sum of the first 10 terms of the series 4 11 + 4 1^4 + 4 21 + 4 2^4 + 4 31 + 4 3^4 + is mn, where (m, n) = 1, then m + n is equal to

The sum 1 + 3 + 11 + 25 + 45 + 71 + upto 20 terms, is equal to

The sum 1 + 1 + 32! + 1 + 3 + 53! + 1 + 3 + 5 + 74! + upto terms, is equal to

1 + 3 + 5^2 + 7 + 9^2 + upto 40 terms is equal to

The sum of the infinite series ^-1 ( 74) + ^-1 ( 194) + ^-1 ( 394) + ^-1 ( 674) + is:

If the sum of the first 20 terms of the series 4 14 + 3 1^2 + 1^4 + 4 24 + 3 2^2 + 2^4 + 4 34 + 3 3^2 + 3^4 + 4 44 + 3 4^2 + 4^4 + is mn,…

If is a root of the equation x^2 + x + 1 = 0 and k=1^n ( ^k + 1 ^k)^2 = 20, then n is equal to

If 11^4 + 12^4 + 13^4 + = ^490, 11^4 + 13^4 + 15^4 + =, 12^4 + 14^4 + 16^4 + =, then is equal to:

If 8 = 3 + 14(3 + p) + 14^2(3 + 2p) + 14^3(3 + 3p) +, then the value of p is:

If and are the roots of x^2 - x - 1 = 0, and Sn = 2023 ^n + 2024 ^n, then:

Let = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 +... up to 10 terms and = n=1^10 n^4. If 4 - = 55k + 40, then k is equal to:

The sum of the series: ( 11 - 3 (1^2) + 1^4) + ( 21 - 3 (2^2) + 2^4) + ( 31 - 3 (3^2) + 3^4) + up to 10 terms

Evaluate 1 2^2 + 2 3^2 + + 100 (101)^2 / (1^2 2^2 + 2^2 3^2 + + 100^2 101):

Let = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + up to 10 terms and = n=1^10 n^4. If 4 - = 55k + 40, then k is equal to:

Let = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + up to 10 terms and = n=1^10 n^4. If 4 - = 55k + 40, then k is equal to:

Let ABC be an equilateral triangle. A new triangle is formed by joining the midpoints of all sides of the triangle ABC, and the same…

If ( 1 + 1 + 1 +2 + + 1 +1012) - ( 12 1 + 14 3 + 16 5 + + 12024 2023) = 12024, then is equal to:

Let f(x) be a function such that f(x + y) = f(x)f(y) for all x, y N. If f(1) = 3 and k=1^n f(k) = 3279, then the value of n is:

If 1^3 + 2^3 + 3^3 + ( up to n terms)1 3 + 2 5 + 3 7 + ( up to n terms) = 95, then the value of n is:

Consider a function f: N R satisfying f(1) + 2f(2) + 3f(3) + + x f(x) = x(x + 1)f(x) for x 2, with f(1) = 1. Then f(2022) is equal to:

Let a1 = b1 = 1 and an = an-1 + (n-1), bn = bn-1 + an-1, n 2. If S = n=1^ an2^n and T = n=1^ bn2^n, then 2^7(2S - T) is equal to:

If an = -24n^2 - 16n + 15, then a1 + a2 + + a5 is equal to:

Let n=0^ n^3 ( (2n)!) + (2n-1)(n!)(n!)(2n)! = a e + be + c, where a, b, c Z and e = n=0^ 1n!. Then a^2 - b + c is equal to.

Let S be the set of all values of a1 for which the mean deviation about the mean of 100 consecutive positive integers a1, a2, a3,, a100 is…

The sum 1^2 - 2 3^2 + 3.5^2 - 4.7^2 + 5.9^2 - + 15.29^2, is:

The sum to 10 terms of the series 11+1^2+1^4 + 21+2^2+2^4 + 31+3^2+3^4 + is:

The sum n=1^ 2n^2 + 3n + 4(2n)! is equal to:

The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + is:

JEE Main 2023 Questions with Solutions Mathematics Section – A If gcd (m, n) = 1 and 1^2 - 2^2 + 3^2 - 4^2 + + (2022)^2 - (2023)^2 = 1012…

If (20)^19 + 2 (21) (20)^18 + 3 (21)^2 (20)^17 + + 20 (21)^19 = k (20)^19, then k is equal to:

Let SK = 1 + 2 + + KK and j=1^n Sj^2 where A, B, C, D N and A has the least value. Then:

If an = n^3n^4 + 147, n = 1, 2, 3,, and an is the greatest term in the sequence, then an is equal to:

If Sn = 4 + 11 + 21 + 34 + 50 + to n terms, then 160(S29 - S9) is equal to:

For k N, if the sum of the series 1 + 4k + 8k^2 + 13k^3 + 19k^4 + is 10, then the value of k is.

Let < an be a sequence such that a1 + a2 + + an = n^2 + 3n(n+1)(n+2). If k=1^10 1ak = p1p2p3 pm where p1, p2,, pm are the first m prime…

Section – B The sum to 20 terms of the series 2.2^2 - 3^2 + 2.4^2 - 5^2 + 2.6^2 - is equal to --.

Let f(x) = k=1^ x^k, where x R and f(2) = 119. Then f(2) - f(1) is equal to.

Let [ ] denote the greatest integer. Then [ 1] + [ 2] + [ 3] + + [ 20] is equal to

If the sum of the series ( 12- 13) + ( 12^2- 12 3+ 13^2) + ( 12^3- 12^2 3+ 12 3^2- 13^3) + ( 12^4- 12^3 3+ 12^2 3^2- 12 3^3+ 13^4) + is,…