Find the indicated sum

Question

Find the indicated sum

(∑^20 n=0) 5(5/3)ⁿ=

Answer 1

∑{n=0,20}(5(5/3)^n)

= 5( 1 + 5/3 + (5/3)^2 + ... + (5/3)^19 + (5/3)^20)

 

S = 1 + 5/3 + (5/3)^2 + ... + (5/3)^19 + (5/3)^20

 

(5/3)S = 5/3 + (5/3)^2 + ... + (5/3)^20 + (5/3)^21

 

(5/3)S - S = (5/3)^21 - 1

 

(5/3 - 1)S = (5/3)^21 - 1

 

S = ((5/3)^21 - 1)/(5/3 - 1)

 

∑{n=0,20}(5(5/3)^n) = 5S = 5((5/3)^21 - 1)/(5/3 - 1)

 

= 5((5/3)^21 - 1)/(2/3) = (15/2) ((5/3)^21 - 1)

 

≈ 341881.403473919475