13. Show that 9v1 - 8 - 9 is divisible by 64, whenever n is a positive

1.	13. Show that 9v1 - 8 - 9 is divisible by 64, whenever n is a positive integer.
Answer» Use mathematical induction method to solve it. it's very easy Use mathematical induction method to solve it. it's very easy Solution: n = 1 ⇒ 9^(n + 1) - 8n - 9 = 9² - 8 - 9 = 81 - 17 = 64=1 x (64) n = 2 ⇒9^(n + 1 )- 8n - 9 = 9³ - 8(2) - 9 = 729 – 16 - 9 = 704= 11 x (64) From n = 3, 4, 5,....9^(n + 1) – 8n - 9 = 9^{(1 + 8)n }- 8n - 9 = 9 [nC0 + nC1 . 8 + nC2.8²+ ... nCn x 8^n] – 8n - 9 = 9[1 + 8^n + nC2.8²+ ... nCn x 8^n] –8n – 9 = 9 + 72n + 9. nC2. 8² + ... 9 x nCn x 8^n –8n - 9 = 8² [n + 9 (nC2 + nC3.8 +... nCn 8^(n-2))] which is divisible by 64.

13. Show that 9v1 - 8 - 9 is divisible by 64, whenever n is a positive integer.

Answer»

Use mathematical induction method to solve it. it's very easy

Solution: n = 1 ⇒ 9^(n + 1) - 8n - 9 = 9² - 8 - 9 = 81 - 17 = 64=1 x (64)

n = 2 ⇒9^(n + 1 )- 8n - 9 = 9³ - 8(2) - 9 = 729 – 16 - 9 = 704= 11 x (64)

From n = 3, 4, 5,....9^(n + 1) – 8n - 9 = 9^{(1 + 8)n }- 8n - 9

= 9 [nC0 + nC1 . 8 + nC2.8²+ ... nCn x 8^n] – 8n - 9

= 9[1 + 8^n + nC2.8²+ ... nCn x 8^n] –8n – 9

= 9 + 72n + 9. nC2. 8² + ... 9 x nCn x 8^n –8n - 9

= 8² [n + 9 (nC2 + nC3.8 +... nCn 8^(n-2))]

which is divisible by 64.