Question

Observe the following pattern
$\left(1\times 2\right)+\left(2\times 3\right)=\frac{2\times 3\times 4}{3}$
$\left(1\times 2\right)+\left(2\times 3\right)+\left(3\times 4\right)=\frac{3\times 4\times 5}{3}$
$\left(1\times 2\right)+\left(2\times 3\right)+\left(3\times 4\right)+\left(4\times 5\right)=\frac{4\times 5\times 6}{3}$
and find the value of
(1 × 2) + (2 × 3) + (3 × 4) + (4 × 5) + (5 × 6)

Answer 1

The RHS of the three equalities is a fraction whose numerator is the multiplication of three consecutive numbers and whose denominator is 3.
If the biggest number (factor) on the LHS is 3, the multiplication of the three numbers on the RHS begins with 2.
If the biggest number (factor) on the LHS is 4, the multiplication of the three numbers on the RHS begins with 3.
If the biggest number (factor) on the LHS is 5, the multiplication of the three numbers on the RHS begins with 4.
Using this pattern, (1 x 2) + (2 x 3) + (3 x 4) + (4 x 5) + (5 x 6) has 6 as the biggest number. Hence, the multiplication of the three numbers on the RHS will begin with 5. Finally, we have:

$1\times 2+2\times 3+3\times 4+4\times 5+5\times 6=\frac{5\times 6\times 7}{3}=70$