Question

Find the nth term and the sum to n terms of the series for $1+5+13+29+61+....$

• A. $T_n=2^{n+1}-3$
  $S_n=(2^2+2^3+2^{n+1})-3n$
• B. $T_n=2^{n+1}+3$
  $S_n=(2^2+2^3+2^{n+1})+3n$
• C. $T_n=2^{n+1}-3$
  $S_n=(2^2+2^3-2^{n+1})-3n$
• D. $T_n=2^{n-1}+3$
  $S_n=(2^2+2^3+2^{n-1})+3n$

Answer 1

The correct option is A $T_n=2^{n+1}-3$

$S_n=(2^2+2^3+2^{n+1})-3n$

$S=1+5+13+29+61+T_n$

$S=1+5+13+29+........T_n$

$0=1+4+8+16+32-T_n$

$4+8+16+\mathrm{32.......}$ sum of GP with $r=2$ and $a=4$

$T_n=1+4(2^{n-1}-1)=2^{n+1}-3$

$S_n=\sum (2^{n+1}-3)=(2^2+2^3+2^{n+1})-3n$

$=2^2\left(\frac{2^n-1}{2-1}\right)-3n=2^{n+2}-4-3n$