Firstly note that ${ S }_{ n }-{ S }_{ n-1 }={ a }_{ n }$

First we write ${ a }_{ n+1 }={ S }_{ n }+n+1$

Then we write ${ a }_{ n }={ S }_{ n-1 }+n$

Subtract the equations to get :

${ a }_{ n+1 }=2{ a }_{ n }+1$

Put in some values to get ${ a }_{ 1 }=1,{ a }_{ 2 }=3,{ a }_{ 3 }=7,{ a }_{ 4 }=15$

The pattern is obvious ${ a }_{ n }={ 2 }^{ n }-1$

Put $n=10$ to get $\boxed { { a }_{ 10 }=1023 }$

just compile the below c program u'll get the answer right away

include<stdio.h>

void main() { int an=1,sn=1,i,n=9; scanf("%d",&n); for(i=1;i<=n;i++) { an=sn+i+1; sn=an+sn; } printf("%d\n",an); }