Finding a1 in an infinite geometric sequence

[Emily_LKJ]

Hi, i really cant seem to figure this question out.
I have the sum of the 5 first numbers of the sequence = 220 and then i have the sum of the whole infinite geometric sequence = 194.4
The two question i have is first
1. How do i show this information as an equation system.
5 first numbers of the sequence = 220
sum of the whole infinite geometric sequence = 194.4

2. How do i find a1 with this as my information (5 first numbers of the sequence = 220, sum of the whole infinite geometric sequence = 194.4)

3. How do i find ratio with this information (5 first numbers of the sequence = 220, sum of the whole infinite geometric sequence = 194.4)

 

[HallsofIvy]

A geometric series (sum of a geometric sequence) is of the form \(\displaystyle a+ ar+ ar^2+ \cdot\cdot\cdot+ ar^n+ \cdot\cdot\cdot\). If you are asked to do this problem the you are expected to have learned that the sum of a finite geometric series, with n terms, is \(\displaystyle a+ ar+ \cdot\cdot\dot+ ar^n= \frac{a(1- r^{n+1})}{1- r}\) while the sum of an infinite geometric series (which only converges if |r|< 1) is \(\displaystyle \frac{a}{1- r}\).

If the first 5 terms sum to 220 then we must have \(\displaystyle a+ ar+ ar^2+ ar^3+ ar^4+ ar^5= \frac{a(1- r^6)}{1- r}= 220\). If the entire infinite series sums to 199.4 then we must have \(\displaystyle \frac{a}{1- r}= 199.4\). Solve those two equations for a and r.

(I would start by dividing the first equation by the second. That will eliminate a giving \(\displaystyle \frac{1- r^6}{1- r}= \frac{199.4}{220}\).)