# If a+b+...+l is a GP, prove that its sum is bl-a^2/b-a

given: a,b, ........I are in GP.
therefore
common ratio r = b/a
let the number of terms in GP be n.
therefore the sum of n terms of GP is $\frac{a\left({r}^{n}-1\right)}{r-1}$ [since a is the first term of GP]
nth term of GP is $I=a.{r}^{n-1}⇒{r}^{n-1}=\frac{I}{a}$

sum $=\frac{a\left(\frac{bI}{{a}^{2}}-1\right)}{\frac{b}{a}-1}=\frac{a.\left(\frac{bI-{a}^{2}}{{a}^{2}}\right)}{\frac{b-a}{a}}$
$=\frac{bI-{a}^{2}}{b-a}$

hope this helps you

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