2 Find closedform formulas for the following recurrence rela

2. Find closed-form formulas for the following recurrence relations a. ao 5, and an n-1 3 or n 2 1 b. ao 3, and an am-1 t n. or n 1 for n 21 c. ao 7, and an 8am-1 d. ao 1, and an an-1 for n 2 1

Solution

(a)

Given a_n=2a_n-1-3

so a_n = 4a_n-2-6-3......

a_n = 2ⁿa₀-3-6-12-... = 2ⁿa_o-3(2^n-1-1)=5*2ⁿ-3*2^n-1+3= 7*2^n-1+3

or a_n = 7*2^n-1+3

(b)

Given a_n = a_n-1+n

=>so on adding,a₁ = a₀+1

a₂=a₁+2

.

.

..

a_n=a_n-1+n

=>a_n = a₀+(1+2+3+....+n )

=>a_n = -3+(n(n+1))/2 = (n²+n-6)/2

(c)

Given a_n = 8a_n-1

so a₁/a₀ = 8, a₂/a₁ = 8 , a₃/a₂ = 8,.....a_n/a_n-1 = 8

on multiplying, a_n = 8ⁿa₀ = 7*8ⁿ

(d)

Given a_n = 6a_n-1/n²

i.e, a₁/a₀ = 6/1², a₂/a₁ = 6/2², a₃/a₂ = 6/3², .....a_n/a_n-1 = 6/n²

on multiplying all, we get a_n = 6ⁿ/(n!)² a₀

=>a_n = (6/n!)²