Question

20. 102n-1is divisible by 11

Answer 1

The given statement is,

P( n ): 10 2n−1 +1 is a multiple of 11(1)

Here, n is a natural number.

For n=1, the statement is,

P( 1 ):( 10 2( 1 )−1 +1 ) is a multiple of 11 P( 1 ):( 10 2−1 +1 ) is a multiple of 11 P( 1 ):11 is a multiple of 11

Hence, P( 1 ) is true.

Consider, n=k(here, k is some positive integer). Substitute n=k in the statement in equation (1).

P( k ): 10 2k−1 +1 is a multiple of 11(2)

According to the principle of mathematical induction, assume that the statement P( n ) is true for n=k. If it is also true for n=k+1, then P( n ) is true for all natural numbers.

Substitute n=k+1 in the statement in equation (1).

P( k+1 ): 10 2( k+1 )−1 +1 is a multiple of 11 P( k+1 ): 10 2k+2−1 +1 is a multiple of 11 P( k+1 ): 10 2k+1 +1 is a multiple of 11 P( k+1 ): 10 2k+1+1−1 +1 is a multiple of 11 (3)

Further analyze from statement (3).

P( k+1 ): 10 2k−1+2 +1 is a multiple of 11 P( k+1 ): 10 2k−1+2 +100−100+1 is a multiple of 11 P( k+1 ):100( 10 2k−1 +1 )−99 is a multiple of 11 P( k+1 ):100( 10 2k−1 +1 )−9( 11 ) is a multiple of 11 (4)

Thus, if P( k ) is true, then P( k+1 ) is also true.

Hence, statement P( n ) is true.