Question

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

Solution

## 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:$$(1\times 2)+(2\times 3)+(3\times 4)+(4\times 5)+(5\times 6)=\dfrac{5\times 6\times 7}{3}=70$$Mathematics

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