Multiplication Identities - STD 9 - Page 66.1.3 EM

Difference of Products - Six Consecutive Numbers

Difference of Products

Six consecutive natural numbers

Problem

Find the difference of the products of the two numbers at the ends and the two numbers in the middle of six consecutive natural numbers.

Six Consecutive Natural Numbers
Product of two numbers at the ends
n × (n+5)
= n(n+5)
Product of two numbers in the middle
(n+2) × (n+3)
= (n+2)(n+3)
Difference
Ends Product − Middle Product
= -6

Algebraic Proof

Let the six consecutive numbers be: n, n+1, n+2, n+3, n+4, n+5

Product at ends = n × (n+5) = n² + 5n

Product in middle = (n+2) × (n+3) = n² + 5n + 6

Difference = (n² + 5n) − (n² + 5n + 6) = -6