- Home
- Standard 11
- Mathematics
8. Sequences and Series
medium
If the sum of $n$ terms of a $G.P.$ is $255$ and ${n^{th}}$ terms is $128$ and common ratio is $2$, then first term will be
A
$1$
B
$3$
C
$7$
D
None of these
Solution
(a) Given that $\frac{{a({r^n} – 1)}}{{r – 1}} = 255$ $(\because \;\;r > 1)$ …..$(i)$
$a{r^{n – 1}} = 128$ …..$(ii)$
and common ratio $r = 2$ …..$(iii)$
From $(iii), (i)$ and $(ii)$
we get $a{2^{n – 1}} = 128$ …..$(iv)$
and $\frac{{a({2^n} – 1)}}{{2 – 1}} = 255$ …..$(v)$
Dividing $(v)$ by $(iv)$
we get $\frac{{{2^n} – 1}}{{{2^{n – 1}}}} = \frac{{255}}{{128}}$
$ \Rightarrow $$2 – {2^{ – n + 1}} = \frac{{255}}{{128}}$
$ \Rightarrow $${2^{ – n}} = {2^{ – 8}}$
$ \Rightarrow $$n = 8$
Putting $n = 8$ in equation $(iv),$
we have $a\;.\;{2^7} = 128 = {2^7}$or $a = 1$.
Standard 11
Mathematics
