Using Binomial theoram, prove that 23n - 7n -1 is divisible by 49 where n is a Natural number

  • 6

p(k) : 23k - 7k - 1= 49 m [ where m is any integer]

or 2 3k = 49 m + 7k + 1

p(k+1) ; 23(k+1)- 7(k+1) - 1

=2 3k+3 - 7(k+ 1) -1

= (49 m + 7k + 1) x 23 - 7(k+ 1) -1

= 49 m 23 + 7 k 23 + 23 - 7k - 7 -1

= 49 m 23 +7k ( 8 -1) + 8 - 8

= 49 (8m + k )

thus divisible by 49

  • -2

Here it shouldn't be solve by using M.I.......... I want to solve it by using binomial like

as we know 23n -7n -1 = 49 k

now put 23n = (1+1)3n and then expand it and prove it

(1+1)3n = 3nC0(1)3n(1)0 +3nC1(1)3n-1(1)1 +.........+3nC3n(1)3n

Hence solve this and prove the above thing.

  • -31
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