Question

Find r for the geometric series

Answer 1

an = a5 = 324
a1 = 4
n = 5
Solve for r

Answer 2

For a geometric series, you have \[\large a_{n}=a_{1}r^{n-1}\]\[\large a_{n}=324 \ , \ a_{1}= 4 \ ,\ n= 5\] plug it all in to the formula.\[\large 324 = (4)r^{5-1}\]divide both sides by 4.\[\large 81 = r^4\] take the fourth root of r on both sides. \[\large \sqrt[4]{81}= \sqrt[4]{r^4}\]\[\large r=3\]

Answer 3

why not \(\large r = -3\)

Answer 4

Sorry, \[\large r= \pm3\]

Answer 5

this is like the same problem we did earlier right?

Answer 6

yes but sum is greater than \(a_5\) here, so you need to resolve it bit differently...

Answer 7

Now to check to see if the ratio is 3, plug it back into the Geometric Sum. \[\large S_{n}= a_{1}\left(\frac{1-r^n}{1-r}\right)\]\[\large 484=4\left(\frac{1-3^5}{1-3}\right)\]\[\large 484 = 484 \]

Answer 8

\(r=-3\) would not work in this case becasue it wouldnt equal our sum, so therefoer we can only use r=+3