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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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