Question

Find the next two terms in the sequence:
$1,3,7,15,31,.......$

Answer 1

$t_1=1,t_2=3,t_3=7,t_4=15,t_5=31$
$t_2=3=2t_1+1$
$t_3=2t_2+1$
$t_4=15=2t_3+1$
$\therefore$ $t_n=2t_{n-1}+1$
$t_6=2t_5+1=2\times 31+1=2\times 31+1=62+1=63$
$t_7=2t_6+1=2\times 63+1=127$
$\therefore$ The next two terms of the sequence are $63,127$.