Question

If the tenth term of the sequence $S=1+5+13+29+.\dots \dots .$ is $k,$ then $k$ is equal to

Answer 1

Answer: 2045

$S=1+5+13+29+.\dots \dots +t_9+t_{10}$
$S=.\dots \dots 1+5+13+.\dots \dots .+t_8+t_9+t_{10}$
Subtracting we get $0=1+\left(4+8+16+.\dots \dots 9terms\right)-t_{10}$
$\Rightarrow t_{10}=1+\left(4+8+16+.\dots ..9terms\right)=1+4\frac{\left(1-2^9\right)}{\left(1-2\right)}$
$=1+4\left(2^9-1\right)=1+4\left(512-1\right)$
$=2045$
$\Rightarrow k=2045$ .