Formula for the sum of a geometric series

Question

I'm using a book for my AS Level maths which says that:

"The general rule for the sum of a geometric series is $$S_n = a\frac{r^n-1}{r-1}$$ or $$S_n= a\frac{1-r^n}{1-r}$$ "

Why are there two formulas and when do I use each?

Thanks so much, I'm really confused!

Sami x

They are the same ; multiply numerator and denominator by $-1$ to pass from one to the other — marwalix, Apr 26 '15 at 16:39
The general rule only applies if $r \neq 1$. If $r = 1$, then $S_n = na$. — N. F. Taussig, Apr 26 '15 at 20:14

score 4 · Accepted Answer · answered Apr 26 '15 at 16:38

4

They are the same formula, the only difference is that you take out a $-1$ at the numerator and at the denominator so they turn into $1$ which is obv not written. Let me show you:

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

answered Apr 26 '15 at 16:38

AlienRem

4,011

Sorry if i'm being really dim- I'm trying to teach myself and can't always manage it! – Sami Apr 26 '15 at 16:42
Vote mine as answer if you find it complete, btw is always a pleasure to help. – AlienRem Apr 26 '15 at 16:58

Formula for the sum of a geometric series

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