Question

13. Show that 9#l- 8n 9 is divisible by 64, whenever n is a positive integer.

Answer 1

The given expression 9 n+1 −8n−9 we have to show that it is divisible 64 where n is a positive integer value given in question.

For prove of divisibility of expression,

9 n+1 −8n−9=64k , where k is some natural number.

Using Binomial Theorem,

( 1+a ) m = C m 0 + C m 1 a+ C m 2 a 2 +..........+ C m m a m

To get it in binomial form we have to split 9 n+1 into ( 1+8 ) n+1 .

So, by comparing both the expression ( 1+a ) m and ( 1+8 ) n+1 , values of a=8 and m=n+1

( 1+8 ) n+1 = C n+1 0 + C n+1 1 ( 8 )+ C n+1 2 ( 8 ) 2 +..........+ C n+1 n+1 ( 8 ) n+1 =1+( n+1 )( 8 )+ 8 2 [ C n+1 2 + C n+1 3 ( 8 )+..........+ C n+1 n+1 ( 8 ) n−1 ] =9+8n+64[ C n+1 2 + C n+1 3 ( 8 )+..........+ C n+1 n+1 ( 8 ) n−1 ]

9 n+1 −8n−9=64k , where k= C n+1 2 + C n+1 3 ( 8 )+..........+ C n+1 n+1 ( 8 ) n−1 is a natural number

Thus 9 n+1 −8n−9 is divisible by 64 , whenever n is positive integer.