Question

Find sum of the first $10$ terms of the series:
$\left(1\right)\left(5\right)+\left(2\right)\left(6\right)+\left(3\right)\left(7\right)+\left(4\right)\left(8\right)+....$

• A. $506$
• B. $605$
• C. $572$
• D. $563$

Answer 1

The correct option is B $605$

We have to find the value of $1\times 4+2\times 6+3\times 7+4\times 8+............$ , upto $10$ terms

$\Rightarrow 1\times 5+2\times 6+3\times 7+4\times 8+........=1(1+4)+2(2+4)+3(3+4)+4(4+4)+.......$

$\Rightarrow {(1}^2+2^2+....+{10}^2)+4(1+2+...10)$

$\Rightarrow \frac{10\times 11\times 21}{6}+4\left(\frac{10\times 11}{2}\right)=605$

Therefore the option is $B$