# 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)$

Given:

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

To do:

We have to find the value of $(1 \times 2)+(2 \times 3)+(3 \times 4)+(4 \times 5)+(5 \times 6)$.

Solution:

We observe that,

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

$(1 \times 2)+(2 \times 3)+(3 \times 4)+(4 \times 5)+(5 \times 6)=\frac{5\times6\times7}{3}$

$=5\times2\times7$

$=70$

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Updated on: 10-Oct-2022

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