solve the recurrence relation a1=2 and an=an-1+2^n for all n>=2

Question

Find associate linear homogenous recurrence relation ,Solution an(h) to the associated linear homogenous recurrence relation ,Find particular solution an(p) to the original recurrence relation by assuming that it must be of the form an(p)=@2^n, Add answers above find the solution ,Give form of the solution to the linear homogineous recurrence relation whose characteristic polynomial is +r-4)^5( r-5)^4

Answer 1

a_n=a_n-1+2ⁿ; a₁=2,

a₂=a₁+4=6; a₃=a₂+8=14; a₄=a₃+16=30;...

Series=2(1), 2(3), 2(7), ..., 2(2^n-1)

a_n=2(2ⁿ-1) or 2ⁿ⁺¹-2.

General form for (r-4)⁵(r-5)⁴:

a_n=a₀(4ⁿ)+a₁n(4ⁿ)+a₂n²(4ⁿ)+a₃n³(4ⁿ)+a₄n⁴(4ⁿ)+b₀(5ⁿ)+b₁n(5ⁿ)+b₂n²(5ⁿ)+b₃n³(5ⁿ), where a_i and b_i are constant coefficients.