Find the 5th partial sum

Question

For the sequence A_n=(2/3ⁿ),

its fifth partial sum S₅=

its nth partial sum S_n=

 

Answer 1

An=(2/3^n)

Sn = 2/3 + 2/3^2 + ... + 2/3^n

(2/3) Sn = 2/3^2 + ... + 2/3^n + 2/3^(n+1)

(2/3) Sn - Sn = 2/3^(n+1) - 2/3

Sn = (2/3^(n+1) - 2/3) / (2/3 - 1)

Sn = (2/3)(1/3^n - 1) / (–1/3)

Sn = –3(2/3)(1/3^n - 1)

Sn = 2 – 2/3^n

 

[Note: n → ∞ => Sn → 2]

S5 = 2 – 2/3^5
= 2(3^5 – 1)/3^5
= 2*242/243
= 484/243