Question

Using Mathematical Induction
to
show that
1 + 2 + 2² +.
+2n = 2n+¹ - 1. (n is power of 2 in all question)​

Answer 1

Answer:

1+2+2²+2³+...+2^n = 2^(n+1)-1

Given :

1+2+2²+2³+...+2^n

To find :

Show that :1+2+2²+2³+...+2^n = 2^(n+1)-1

Solution :-

On taking LHS

Let P(n) = 1+2+2²+2³+...+2^n

Put n = 0 then

Since given problem for non negtive integers we are starting with 0

P(1) = 2⁰= 1

RHS = 2^(0+1)-1

=> 2¹-1

=> 2-1

=> 1

LHS = RHS

Now

Let For n , P(n) be true

P(n)=2^(n+1)-1———(i)

Now we compute P(n+1)

P(n+1)=1+2+2²……..+2^n+2^(n+1)

=> P(n+1) = P(n)+2^(n+1)

From (i) we have

P(n+1)=2^(n+1)+2^(n+1)-1

=> P(n+1) = 2×(2^(n+1)) -1

=> P(n+1) = 2^(n+1+1)-1

=> P(n+1) = 2^(n+2) -1

Now ,

Putting n=n+1 in RHS

=> 2^(n+1+1)-1

=> 2^(n+2) -1

LHS = RHS

Therefore, P(n+1) is true.

Therefore expression is true for all

Hence, Proved.

Used formulae:-

Induction Method :-

In this method we assumes for n and then uses this assumption to prove that the statement holds for n + 1.