Question

CAN ANYONE SOLVE THIS PROBLEM IMMEDIATELY......


Let P (n) be the statement "7 divides (2^(3n) -1)". what is P (n+1)?​

Answer 1

Answer:

P ( n ) = 2^(3n) - 1 is divisible by 7.

P (1) = 8 - 1 = 7

P(1) is true.

Let us assume that P( k ) is true for some positive integer.

P( k) = 2^ (3k) - 1 = 7m

P ( k ) = 2 ^3k = 7m +1 --> ( i )

Now, we shall prove that P ( k +1 ) is also true.

P ( k +1) = 2 ^3( k + 1) - 1

P ( k + 1) = 2 ^( 3k + 3 ) - 1

P ( k +1 ) = ( 2^3k ) 2^3 - 1

P ( k + 1) = ( 7m + 1) 2^3 - 1

P ( k +1) = ( 7m + 1). 8 - 1

P ( k +1) = 56 m + 8 - 1 = 56m + 7

P ( k +1) = 7 ( 8m + 1)

Let 8m + 1 = lambda

So,

P ( k +1) = 7 lambda

That means P ( k +1) is also divisible by 7.

♥️Hence, P ( k +1) or P ( n +1) is also true.