Question

if $\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right).....\left(1+x^{128}\right)=\sum _{r=0}^n{x}^r$ then find the value of $n$

Answer 1

Consider the given series

$(1+x)(1+x^2)(1+x^4).....(1+x^{128})={{\sum }^n}_{r=0}{x}^r$

Where $r=0$ and limit is $n$

Take degree of polynomial on $L.H.S.$

Then, $1+2+4+.............+128$

This is G.P. series Now,

$firstterma=1$

$commonratior=\frac{2}{1}=2$

$numberofterms=8$

Then Sum of series $Sum=\frac{a(r^n-1)}{(r-1)}when,r>1$

$S=\frac{1(2^8-1)}{(2-1)}$

$\Rightarrow n=S=256-1$

$\Rightarrow n=255$

Hence, this is the answer