geometric sequences

[popeyesmom]

Hello. I have been a 6th grade math teacher for 25 years. I have not been to school myself in 25 years, many workshops, but not school. I please need help with a math problem explained, if possible, on a middle school level. 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 this sequence?

 

[pka]

Hello. I have been a 6th grade math teacher for 25 years. I have not been to school myself in 25 years, many workshops, but not school. I please need help with a math problem explained, if possible, on a middle school level. 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 this sequence?

The first three terms are \(\displaystyle a,~ar,~\&~ar^2\).

Thus \(\displaystyle a+ar+ar^2=7a\) and \(\displaystyle a+ar+ar^2+ar^3=45\).

 

[Novice]

I think she knows this much but cannot go beyond it. So I think we should help.
Get the value of r from the first equation ( a will cancel out, leaving you with a quadratic in r, which should be positive and greater than 1 because the series is of positive numbers, and it is increasing as the sum of first three numbers is 7 times the first ). Putting this value of r in the second equation, you should be able to find a.