Determine the number of terms \(n\) in the geometric series.

a1 = 320; r = 1/4, Sn = 6825/16

a) 9

b) 7

c) 6

d) 5

Guest Jun 4, 2021

#1**+1 **

I just did a similar one to this, only with an AP. Have a go by yourself.

Show us what you try, even if you don't get far.

Melody Jun 4, 2021

#2**+1 **

I've already tried. I got to 455/256 = 1/4^n but when I converted it into log and inputted it into my calculator, it said it couldn't do the problem. I also got -181999 = 1/4^n but that also wouldn't work when I tried getting the final answer through converting it into log. I hope this makes sense. I was using log to simplify for n in both instances. That's the only way I know how to do it. I'm not even sure if 455/256 and -181999 are the right end numbers. I'm confused and I need help! I hope I've shown you that I tried. Along the way, I also tried different ways of simplifying and canceling out too. I'm stuck.

Guest Jun 4, 2021

#5**+1 **

this is a geometric progression problem

a1 = 320; r = 1/4, Sn = 6825/16 find n

For a GP

\(S_n=\frac{a(1-r^n)}{1-r}\\ a=320, \quad r=0.25\quad S_n=6825/16 = 426.5625\\~\\ \frac{6825}{16}=\frac{320(1-0.25^n)}{1-0.25}\\ \frac{6825}{16}=\frac{320(1-0.25^n)}{0.75}\\ \frac{6825}{16}*\frac{3}{4}*\frac{1}{320}=1-0.25^n\\ \frac{1365}{16}*\frac{3}{4}*\frac{1}{64}=1-0.25^n\\ \frac{4095}{4096}=1-0.25^n\\ 0.25^n=1-\frac{4095}{4096}\\ 0.25^n=\frac{1}{4096}\\ \frac{1}{4^n}=\frac{1}{4096}\\ 4^n=4096\\ 4^n=4^6\\ n=6\)

LaTex:

S_n=\frac{a(1-r^n)}{1-r}\\

a=320, \quad r=0.25\quad S_n=6825/16 = 426.5625\\~\\

\frac{6825}{16}=\frac{320(1-0.25^n)}{1-0.25}\\

\frac{6825}{16}=\frac{320(1-0.25^n)}{0.75}\\

\frac{6825}{16}*\frac{3}{4}*\frac{1}{320}=1-0.25^n\\

\frac{1365}{16}*\frac{3}{4}*\frac{1}{64}=1-0.25^n\\

\frac{4095}{4096}=1-0.25^n\\

0.25^n=1-\frac{4095}{4096}\\

0.25^n=\frac{1}{4096}\\

\frac{1}{4^n}=\frac{1}{4096}\\

4^n=4096\\

4^n=4^6\\

n=6

Melody Jun 4, 2021

#6**+1 **

Thank you so much. I wasn't using decimals so that might have made the problem a bit more tricky to follow.

Guest Jun 4, 2021