Write the first 4 terms of the sequence defined by and then solve the following recurrence relation (in Description below).

Question

Write the first 4 terms of the sequence defined by and then solve the following recurrence relation: for n ≥ 1, a_n = 2a_n−1 + 5, where a₀ = 1.

Answer 1

a₀ = 1

a₁ = 2a₀ + 5 = 2 * 1 + 5 = 7

a₂ = 2a₁ + 5 = 2(2a₀ + 5) + 5 = 2²a₀ + 2 * 5 + 5 = 2² * 1 + 2 * 5 + 5 = 19

a₃ = 2a₂ + 5 = 2(2²a₀ + 2 * 5 + 5) + 5 = 2³a₀ + 2² * 5 + 2*5 + 5 = 2³ * 1 + 2² * 5 + 2*5 + 5 = 43

a₄ = 2a₃ + 5 = 2(2³a₀ + 2² * 5 + 2*5 + 5) + 5 = 2⁴a₀ + 2³ * 5 + 2²*5 + 2 * 5 + 5 = 2⁴ * 1+ 2³ * 5 + 2²*5 + 2 * 5 + 5 = 91

Then, an = 2ⁿ a₀ + ∑_i=0^n-1 2^i-1 * 5 = 2ⁿ + 5 * (1 - 2ⁿ)/(1 - 2) = 2ⁿ + 5 * (2ⁿ - 1) = 6 * 2ⁿ - 5