Question

if s1,s2,s3...sp are the sum of infinite geometric series whose first terms are 1,2,3 and whose common ratios are 1/2,1/3,...1/p+1resp prove that S1,S2 S3+...+sp=1/2p(p+3)

Answer 1

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Answer 2

S1+S2 + S3+...........+sp = p(p+3)/2

Step-by-step explanation:

s1,s2,s3...sp are the sum of infinite geometric series

Sum of infinite GP Series S = a/(1-r)

a = 1  r  = 1/2

s1 = 1/(1 - 1/2)  = 2

s2 = 2/(1 - 1/3) = 3

s3 = 3/(1 - 1/4) = 4

s(p-1) = (p-1)/(1 - 1/p)  = p

sp = p/(1 - 1/(p + 1))  = p + 1

2  + 3 +  4 +...................................................p + (p+1)

a = 2

L = p + 1

n = p

Sum = (p/2)(2 + p + 1)   = p(p+3)/2

S1+S2 + S3+...........+sp = p(p+3)/2

QED

Proved

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