Let (P(n): 2+2^2+2^3+ +2^n=2^ n+1 , n N ). Then,

Step-by-step solution

Given, (p(n)=2+2^2+2^3+ .+2^n=2^ n+1 ) Where (n N ). Let (p(m) ) is true then, (P(m)=2+2^2+2^3+ .+2^m=2^ m+1 ) So, ( aligned P(m+1) & =2+2^2+2^3+ .+2^m+2^ m+1 & = (2+2^2+2^3+ .+2^m )+2^ m+1 & =2^ (m+1) +2^ m+1 =2.2^ m+1 =2^ m+2 aligned ) Clearly, if (P(m) ) is true then (p(m+1) ) is also true. ( P(m) ) is true ( P(m+1) ) is true.