Question

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

Answer 1

The R.H.S 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 on the L.H.S is $3,$ the multiplication of the three numbers on R.H.S begins with $2$

If the biggest number on the L.H.S is $4,$ the multiplication of the three numbers on R.H.S begins with $3$.

If the biggest number on the L.H.S is $5,$ the multiplication of the three numbers on R.H.S begins with $4$

Using this pattern, $(1\times 2)+(2\times 3)+(3\times 4)+(4\times 5)+(5\times 6)$ has $6$ as the biggest number.

Hence, the multiplication of the three numbers on the R.H.S will begin with $5.$

Finally, we have:

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