# Show that 9^(n+1) – 8n – 9 is divisible by 64, whenever n is a positive integer.

16.5k views

edited
Show that 9^(n+1) – 8n – 9 is divisible by 64, whenever n is a positive integer.

by (13.6k points)
selected

Solution:
n = 1 ⇒ 9n + 1 - 8n - 9 = 92 - 8 - 9
= 81 - 17 = 64= 1 x (64)
n = 2 ⇒9n + 1 - 8n - 9 = 93 - 8(2) - 9
= 729 – 16 - 9 = 704= 11 x (64)
From n = 3, 4, 5,.....9n + 1 – 8n - 9 = 9(1 + 8)n - 8n - 9
= 9 [nC0 + nC1 . 8 + nC2.82 + ... nCn x 8n] – 8n - 9
= 9[1 + 8n + nC2.82 + ... nCn x 8n] –8n – 9
= 9 + 72n + 9. nC2. 82 + ... 9 x nCn x 8n –8n - 9
= 82 [n + 9 (nC2 + nC3.8 +... nCn 8n-2)]
which is divisible by 64.