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Simple interest and AP

Authors
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    Name
    Vu Hung
    Twitter

Problem Statement

A principal of $2,000,000 is invested at 5.7%5.7\% p.a. simple interest. Let AnA_n be the total amount after nn years.

  • Find A1,A2,A3,A4A_1,A_2,A_3,A_4.
  • Find a formula for AnA_n and evaluate A12A_{12}.
  • How many whole years before the amount exceeds $6,000,000?

Hints

Simple interest adds a fixed amount each year, so (An)(A_n) is arithmetic.

Solutions

Yearly interest is 0.057×2,000,000=114,0000.057\times 2{,}000{,}000=114{,}000. Hence

An=2,000,000+114,000n.A_n=2{,}000{,}000+114{,}000n.

So

A1=2,114,000, A2=2,228,000, A3=2,342,000, A4=2,456,000,A_1=2{,}114{,}000,\ A_2=2{,}228{,}000,\ A_3=2{,}342{,}000,\ A_4=2{,}456{,}000,

and

A12=3,368,000.A_{12}=3{,}368{,}000.

For exceeding $6,000,000:

2,000,000+114,000n>6,000,000  ⟹  n>35.087…2{,}000{,}000+114{,}000n>6{,}000{,}000 \implies n>35.087\ldots

so n=36n=36 years.

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