John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. Explain to John how to represent his first 20 payments in sigma notation. Then explain how to find the sum of his first 20 payments, using complete sentences. Explain why this series is convergent or divergent.
Solution:-
first payment = 100
second payment = 1.1*100 =110
third payment = 1.1*110 = 121
For nth payment = 100*(1.1) (n-1)
first 20 payments in sigma notations
20
S = ∑ 100(1.1) (n-1)
n=1
to find the sum of the first 20 payments we can use the GP sequence sum formula , where a=100, r= 1.1 and n = 20.
This series of the payments is convergent because we can calculate the sum of 20 payment using the GP sum formula , and answer is limited .