Question

Prove that 1ⁿ + 2ⁿ + 3ⁿ + .......... + 15ⁿ is divisible by 480 for all odd n ≥ 5.

Answer 1

Means A is multiple of 32.

So, A + B is also multiple of 32.

Now, (1ⁿ + 2ⁿ) + (4ⁿ + 5ⁿ) + (7ⁿ + 8ⁿ) +........... (13ⁿ + 14ⁿ) + {3ⁿ + 6ⁿ + ........ 15ⁿ}

(xⁿ + yⁿ) is always divisible by (x + y) when n is odd means all factors is divisible by 3.

Now, (1ⁿ + 4ⁿ) + (2ⁿ + 3ⁿ) + (6ⁿ + 9ⁿ) + (7ⁿ + 8ⁿ) + (11ⁿ + 14ⁿ) + (12ⁿ + 13ⁿ) + {5ⁿ + 10ⁿ + 15ⁿ}

(xⁿ + yⁿ) is always divisible by (x + y) when n is odd.

Means all factors is divisible by 5.

Hence given expression is divisible by 32 × 3 × 5 = 480.

Comments

Can you explain which rules are used