If n is any positive integer , show that
` 2^(3n +3) -7n - 8 ` is divisible by 49 .

Explanation

Given expression
` = 2^(3n + 3) - 7n -8= 2^(3n) . 2^(3) - 7n - 8 `
` 8^(n) . 8 - 7n - 8 = 8 (1 + 7)^(n) - 7n - 8`
` = 8 (1 + ""^(n)C_(1) . 7 + ""^(n)C_(2) . 7^(2) + …+ ""^(n)C_(n)_ . 7^(n)) - 7n - 8`
`= 8 + 56n + 8 (""^(n)C_(2) . 7^(2) + ...+ ""^(n)C_(n). 7^(n)) - 7n- 8 `
`= 49 + 8(""^(n)C_(2) . 7^(2) + ... + ""^(n)C_(n) . 7^(n))`
` 49 { n + 8(""^(n)C_(2) + ... + ""^(n)C_(n) . 7^(n-2))} `
Hence , `2^(3n+3) - 7n - 8` is divisible by 49 .