A Proof using Telescopic Sum: $a_1=3,\,2a_n = 1+a_{n-1};\;n\ge 2.$

Question

$ a_1 = 3,\, 2a_n = 1+a_{n-1}; \;n \ge 2 $

Prove that $ a_n = 1 + 2^{2-n};\; n\in Z^+ $ using Telescopic Series.

I was able to do with Mathematical induction, need some help with Telescopic series.

Answer 1

Hint. By multiplying $$ 2a_n = 1+a_{n-1} $$ by $2^{n-1}$ one gets $$ 2^n a_n = 2^{n-1}+2^{n-1} a_{n-1} $$ or $$ 2^n a_n -2^{n-1} a_{n-1}= 2^{n-1} $$ then one has telescoping terms when summing with respect to $n$.