Question

Induction proof for the sum of any five consecutive integers is divisible by 5 (without remainder).

Answer 1

Let five numbers be 1, 2, 3, 4 and 5.

1 + 2 + 3 + 4 + 5 = 15 is exactly divisible by 5.

Let five numbers be k, k + 1, k + 2, k + 3 and k + 4.

Assume that k + k + 1 + k + 2 + k + 3 + k + 4 = 5k + 10 is exactly divisible by 5. (But it's right!)

Let the five numbers be k + 1, k + 2, k + 3, k + 4 and k + 5.

k + 1 + k + 2 + k + 3 + k + 4 + k + 5

=> (k + 1 + k + 2 + k + 3 + k + 4 + k) + 5

=> (k + k + 1 + k + 2 + k + 3 + k + 4) + 5

=> (5k + 10) + 5

We have assume earlier that 5k + 10 is exactly divisible by 5. To this, 5 is added, which is also exactly divisible by 5.

Hence proved by PMI that the sum of any five consecutive integers is exactly divisible by 5.

Answer 2

Hope it helps ! mark as brainliest