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 . Question from glen, a student: A geometric sequence has a first term of 0.1024, a second term of 0.256, and a middle term of 156.25. how many terms are there in the whole sequence? i know that r=2.5 but i don't know how to find how many terms there are.

Hi Glen,

You know that the common ration r is 2.5 since 0.1024 × 2.5 = 0.256. Thus you know the third term is

0.256 × 2.5 = 0.1024 × 2.52 = 0.64

and the fourth term is

0.64 × 2.5 = 0.1025 × 2.53

etc.

What I hope you see here is that each term is 0.1025 × 2.5power where power is one less than the term number. So, for example the fifth term is 0.1025 × 2.55-1 = 0.1025 × 2.54.

Thus if the middle term is term number m then the middle term is 0.1025 × 2.5m-1. Hence

0.1025 × 2.5m-1 = 156.25

so

2.5m-1 = 156.25/0.1025.

Evaluate the right side and then use logarithms to solve for m.

I hope this helps,
Penny

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