Probability .

Question

solve the given reccrence relation problem

x_{n =} x_n-1 + 2ⁿ , x₀ = 5

Answer 1

x_o = 5

x₁= 5+2 = 7

x₂= 7 + 4 = 11

x₃ = 11 + 8 = 19

...

x_n = x_n-1 + 2ⁿ

Answer 2

By substitution

x₁ =5 + 2¹

x₂ = 5 + 2¹ + 2²

x_n = 5 + (2¹ + ... + 2ⁿ)

= 5 + 2 ( 1 + 2 + ... + 2^n-1)

Inside the parentheses we have a geometric series, common ratio 2, n terms.

You can look up a standard result for summing that series