Exams › JEE Advanced › Maths

Let Sₙ be the sum of the first n terms of a sequence {aₙ}. The relation aₙ = 5*Sₙ + 1 holds for all n in N. Define bₙ = (4 + aₙ)/(1 - aₙ) for n in N. Find the value of ((4¹² - 1)*b₁₂ - 1) / 2²⁷.

1. 1/2
2. 1
3. 2
4. 3/2

Correct answer: 1/2

Solution

For n=1: a₁ = 5*a₁ + 1 -> -4*a₁ = 1 -> a₁ = -1/4. For n >= 2: aₙ - aₙ₋₁ = 5*(Sₙ - Sₙ₋₁) = 5*aₙ -> -4*aₙ = aₙ₋₁ -> aₙ = (-1/4)*aₙ₋₁. So {aₙ} is geometric with first term -1/4 and ratio -1/4, giving aₙ = (-1/4)ⁿ. Then a₁₂ = (-1/4)¹² = 1/4¹² (positive). b₁₂ = (4 + 1/4¹²)/(1 - 1/4¹²) = (4*4¹² + 1)/(4¹² - 1). Numerator of expression: (4¹² - 1)*b₁₂ - 1 = (4¹²-1)*(4*4¹²+1)/(4¹²-1) - 1 = 4*4¹² + 1 - 1 = 4¹³. Final value: 4¹³/2²⁷ = 2²⁶/2²⁷ = 1/2.

Related JEE Advanced Maths questions

• If a, b, and c are in harmonic progression, then e raised to the power of -a, e raised to the power of -b, and e raised to the power of -c will be in which progression?
• If x, y, and z represent the pᵗʰ, qᵗʰ, and rᵗʰ terms of both an arithmetic progression and a geometric progression, what is the value of (xʳ)(yᵖ)(zᵠ)?
• Let ϕ(x) represent a quadratic polynomial. Given that ϕ(1) equals ϕ(−1) and the terms a₁, a₂, a₃ form an arithmetic progression, then the values ϕ(a₁), ϕ(a₂), ϕ(a₃) will be in which sequence?
• Let Sₙ = Σ (k+1)/2 * k². Then Sₙ can take value(s)
• If a, b, and c are positive integers such that b is divisible by a, and they form a geometric sequence, while their arithmetic mean equals b + 2, what is the value of (a² + a - 14)/(a + 1)?
• Let bᵢ > 1 for i = 1, 2,..., 101. Assume that logₑb₁, logₑb₂,..., logₑb₁₀₁ form an arithmetic sequence with a common difference of log₂. Also, let a₁, a₂,..., a₁₀₁ form an arithmetic sequence where a₁ = b₁ and a₅₁ = b₅₁. If t represents the sum b₁ + b₂ +... + b₅₁ and s represents the sum a₁ + a₂ +... + a₅₁, then which of the following is true?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →