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 Question from vicki, a teacher: I am a 6th grade math teacher back in school after 25 years of teaching. I have forgotten sooo much. My professor gave me this question that I have no idea how to solve. If you would, please explain in elementary terms. Thank you. The sum of the first three terms of a geometric sequence of positive integers is equal to seven times the first term, and the sum of the first four terms is forty-five. What is the first term of the sequence?

Hi Vicki,

A geometric sequence has the form

$a, ar, ar^2, ar^3, ...$

for some numbers $a$ and $r.$ The fact that "The sum of the first three terms of a geometric sequence of positive integers is equal to seven times the first term" tells you that

$a + ar + ar^2 = 7a.$

You can divide both sides of this equation by $a$ to obtain

$1 + r + r^2 = 7.$

Solve this quadratic for $r.$ There are two solution but you need the one that is positive.

Now take the second fact "and the sum of the first four terms is forty-five" and use it to write an equation. Substitute the value you found for $r$ and solve for $a.$

I hope this helps,
Penny

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