Question

Prove that the sum of any 5 consecutive natural numbers is divisible y5

Answer 1

We can prove it by mathematical induction.

STATEMENT : sum of any 5 consecutive natural numbers is divisible by 5.

let x be a Natural number then, x + 1, x + 2, x + 3, and x + 4 are consecutive numbers of x.

multiples of 5 say, 5y are divisible by 5

so, (x) + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 5y ___(1)

Equation 1 has to be proven,

let us take x = 1,

then, 1 + 2 + 3 + 4 + 5 = 15 = 5×3 which is divisible by 5

so, Equation is true for x = 1

let the equation (1) is true for x = k

then, k + k+1 + k+2 + k+3 + k+4 = 5k+10 = 5z ___(2)

for, x = k+1 equation (1) becomes

k+1 + (k+1)+1 + (k+1)+2 + (k+1)+3 + (k+1)+4 = 5k+15 = 5k+10 + 5 = 5z + 5 = 5 ( z + 1 ) from equation (2)

let z + 1 be a then 5 ( z + 1 ) = 5a which is divisible by 5

So, the statement holds good for x = k+1

so, the given statement is true for all x belongs to Natural numbers.

Hence, proved.