Question

mathematics induction
can anyone solve this question​

Answer 1

Answer:

Let P(n) be the given equation

P(n) : Given question stated above

Put n = 1

Lhs = 2¹ =2

Rhs = 2( 2¹-1)=2 So lhs =rhs Let P ( 1) is true

Put n = k We have P(k) =

2 +2²+2³+....+2^ k= 2( 2^k - 1 ) Equation 1 Let P(k) be true

Put n = k +1

2+2²+2³+....+2^(k+1)= 2 ( 2^(k+1) - 1 )

2+2²+2³+.....+2^( k )×2= 2 ( 2^(k+1) - 1 )

from equation 1 we have

2(2^k-1 ) ×2 = 2( 2^(k +1) -1 )

lhs = rhs

So P( k+1) is true when P ( k) is true

Therefore by PMI P(n) is true for all n belongs to N