Question

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 the amount he paid the month before. Explain to John how to represent his first 20 payments in sequence notation. Then explain how to find the sum of his first 20 payments, using complete sentences.

Answer 1

The sequence can be written as:
p(n)= 100*(1.1)ⁿ⁻¹, from n=1 to n=20

this gives us the value of p, the payment, for every period n
Let's plug in a couple numbers:
n=1 (first payment)
p(1) = 100*(1.1)⁰
P(1) = 100 * 1 = $100 This matches the first payment described

n=3 (third payment)
P(3) = 100*(1.1)³⁻¹
P(3) = 100 * (1.1)²
P(3) = 100 * 1.21 =$121

And so on until n=20 (last payment).
P(20) = 100*1.1¹⁹
P(20) = 100*6.11
P(20) = $612

The summation notation simply adds these payments together, adding each successive nth payment to the total.
n=20
∑(100*(1.1)ⁿ⁻¹)
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