Let \ a _ n \ _ n = 0 ^ ∞ be a sequence such that a _ 0 = a _ 1 = 0 and a _ n + 2 = 3 a _ n + 1 - 2 a _ n + 1, n ≥ 0 . Then a _ 25 a _ 23 - 2 a _ 25 a _ 22 - 2 a _ 23 a _ 24 + 4 a _ 22 a _ 24 is equal to

Correct answer: B. 528 Given, a _ 0 = 0, a _ 1 = 0 And a _ n + 2 = 3 a _ a + 1 - 2 a _ n + 1 : n ≥ 0 ⇒ a _ n + 2 - a _ n + 1 = 2( a _ n + 1 - a _ n ) + 1 Now for n = 0 a _ 2 - a _ 1 = 2 ( a _ 1 - a _ 0 ) + 1 . . . . (1) For n = 1 a _ 3 - a _ 2 = 2( a _ 2 - a _ 1 ) + 1 . . . . . . . . (2) For n = 2 a _ 4 - a _ 3 = 2 ( a _ 3 - a _ 2 ) + 1 . . . . . . . . (3) . . . For n = n a _ n + 2 - a _ n + 1 = 2( a _ n + 1 - a _ n ) + 1 . . . . . . (n) Now adding all above equation upto n we get, ( a _ n + 2 - a _ 1 ) - 2 ( a _ n + 1 - a _ 0 ) - (n + 1) = 0 ⇒ a _ n + 2 = 2 a _ n + 1 + (n + 1) Now replacing n → n - 2 we get, ⇒ a _ n - 2 a _ n - 1 = n - 1 Now putting n = 25 we get, ⇒ a _ 25 - 2 a _ 24 = 25 - 1 = 24 Now putting n = 23 we get, ⇒ a _ 23 - 2 a _ 22 = 23 - 1 = 22 So, ( a _ 25 - 2 a _ 24 ) ( a _ 23 - 2 a _ 22 ) = (24) (22) = 528

JEE Main 202229 JulyEvening Shift

Exam: JEE Main
Year: 2022
Date: 29 July
Shift: Evening Shift
Subject: Mathematics
Chapter: Sequences and Series

JEE Main 2022 · 29 July · Evening Shift · Mathematics · Sequences and Series

Let \ a _ n \ _ n = 0 ^ ∞ be a sequence such that a _ 0 = a _ 1 = 0...

Let ${\left\{{a}_{n}\right\}}_{n = 0}^{\infty}$ be a sequence such that ${a}_{0} = {a}_{1} = 0$ and ${a}_{n + 2} = 3{a}_{n + 1} - 2{a}_{n} + 1, \forall n \ge 0$ .

Then ${a}_{25}{a}_{23} - 2{a}_{25}{a}_{22} - 2{a}_{23}{a}_{24} + 4{a}_{22}{a}_{24}$ is equal to

Options

A $483$
B $528$
C $575$
D $624$

Correct answer

B. $528$

Step-by-step solution

Given,
${a}_{0} = 0, {a}_{1} = 0$
And ${a}_{n + 2} = 3{a}_{a + 1} - 2{a}_{n + 1}: n \ge 0$
$⇒ {a}_{n + 2} - {a}_{n + 1} = 2\left({a}_{n + 1} - {a}_{n}\right) + 1$
Now for $n = 0{a}_{2} - {a}_{1} = 2$ \left({a}_{1} - {a}_{0}\right) + 1 . . . . $\left(1\right)$
For $n = 1{a}_{3} - {a}_{2} = 2\left({a}_{2} - {a}_{1}\right) + 1 . . . . . . . . \left(2\right)For$ n = 2{a}_{4} - {a}_{3} = 2 \left({a}_{3} - {a}_{2}\right) $+ 1 . . . . . . . . \left(3\right)$
. . .
For $n = n{a}_{n + 2} - {a}_{n + 1} = 2\left({a}_{n + 1} - {a}_{n}\right) + 1 . . . . . . \left(n\right)Now adding all above equation upto$ n $we get,\left({a}_{n + 2} - {a}_{1}\right) - 2 \left({a}_{n + 1} - {a}_{0}\right) - \left(n + 1\right) = 0$
$⇒ {a}_{n + 2} = 2{a}_{n + 1} + \left(n + 1\right)Now replacing$ n \to n - 2 $we get,$ ⇒ {a}_{n} - 2{a}_{n - 1} = n - 1 $Now putting$ n = 25 $we get,$ ⇒ {a}_{25} - 2{a}_{24} = 25 - 1 = 24 $Now putting$ n = 23 $we get,$ ⇒ {a}_{23} - 2{a}_{22} = 23 - 1 = 22 $So, \left({a}_{25} - 2{a}_{24}\right) \left({a}_{23} - 2{a}_{22}\right) = \left(24\right) \left(22\right) = 528$

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Let α = 3+4+8+9+13+14+… upto 40 terms. If ( β)^ (α)/(1020) is a root of the equation… JEE Main 2026 The sum 1 + (1)/(2)(1^2 + 2^2) + (1)/(3)(1^2 + 2^2 + 3^2) + … upto 10 terms is equal to : JEE Main 2026 The value of 1^3 - 2^3 + 3^3 - … + 15^3 is: JEE Main 2026 The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a… JEE Main 2026 For the functions f(θ) = α ^2θ + β ^2θ , and g(θ) = α ^2θ + β ^2θ , α > β > 0 , let _ 0 <… JEE Main 2026 If the sum of the first 10 terms of the series (1)/(1 + 1^4 × 4) + (2)/(1 + 2^4 × 4) +… JEE Main 2026 Let A_1, A_2, A_3, …, A_ 39 be 39 arithmetic means between the numbers 59 and 159 . Then… JEE Main 2026 Let the sum of the first n terms of an A.P. be 3n^2 + 5n . Then the sum of squares of the… JEE Main 2026 ∑_ n=1 ^ 10 ( (528)/(n(n+1)(n+2)) ) is equal to: JEE Main 2026 Let α=(1)/(4)+(1)/(8)+(1)/(16)+…∞ and β=(1)/(3)+(1)/(9)+(1)/(27)+…∞ . Then the value of… JEE Main 2026 The first term of an A.P. of 30 non-negative terms is (10)/(3) . If the sum of this A.P.… JEE Main 2026 Let a_1, a_2, a_3, … be an A.P. and g_1 = a_1, g_2, g_3, … be an increasing G.P. If a_1 =… JEE Main 2026 The sum (1^3)/(1) + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ⋯ up to 8… JEE Main 2026 Let α, α + 2, α ∈ ℤ , be the roots of the quadratic equation x(x+2) + (x+1)(x+3) +… JEE Main 2026 Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common… JEE Main 2026 If ∑_ k=1 ^ n a_k = 6n^3 , then ∑_ k=1 ^ 6 ( a_ k+1 - a_k 36 )^2 is equal to _______. JEE Main 2026 Full Sequences and Series list Sequences and Series JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs