Question

John has taken out a loan for college. He started paying off the loan with the first payment of $100. Each month he pays, he wants to pay back 1.1 times the amount he payed the month before. Explain to John how to represent his first 20 payments in sequence notation. Then explain how to find the sum of the first 20 payments using complete sentences.

Answer 1

John's payments are
month 1: $100
month 2: $100*(1.1)
month 3: $100*(1.1)²
month 4: $100*(1.1)³

...

The payments form the geometric sequence
a, ar, ar², ..., arⁿ
where
a = $100
r = 1.1

The sum of the first n terms is

`S_(n) = (a(1-r^(n)))/(1-r)`

Therefore the sum of the first 20 payments is
S₂₀ = [$100(1 - 1.1²⁰)]/(1 - 1.1)
= $5727.50

John makes a total of $5727.50 in the first 20 payments.