Provide formulas for an where a0 1 and an is a an arithmeti

Provide formulas for a_n where a_0 = 1 and a_n is (a) an arithmetic sequence with common difference 4 and (b) a geometric sequence with common ratio 1/4. Compute sigma^infinity k =1 2 times (2/3)^n -1.

Solution

The n the term of arithmetic series is given by

a_n=a₀ + (n-1)d

In the given question,a₀=1 and d=4

therefore

a_n=1 +(n-1)4= 4n-3

The nth term of geometric series is given by

a_n=a₀r^n-1

here r=1/4

a_n=1(1/4)^n-1

a_n=(1/4)^n-1