Question

The sum of $5$ numbers in AP is $30$ and the sum of their squares is $220$. Which of the following is the third term?

Answer 1

Determine the third term:

The sum of $5$ numbers of an A.P is given as $30$ .
Let us take the $5$ numbers as,
$\mathrm{a}-2\mathrm{d},\mathrm{a}-\mathrm{d},\mathrm{a},\mathrm{a}+\mathrm{d},\mathrm{a}+2\mathrm{d}$
Here $\mathrm{a}=$ first term of the A.P and $\mathrm{d}=$ Common difference
So as it is given that their sum is $30$ so,
$\mathrm{a}-2\mathrm{d}+\mathrm{a}-\mathrm{d}+\mathrm{a}+\mathrm{a}+\mathrm{d}+\mathrm{a}+2\mathrm{d}=30$
Take first term value together and common difference value together as follows:
$(\mathrm{a}+\mathrm{a}+\mathrm{a}+\mathrm{a}+\mathrm{a})+(-2\mathrm{d}-\mathrm{d}+\mathrm{d}+2\mathrm{d})=30$
Simplifying further we get,

$\Rightarrow 4\mathrm{a}+0=30$
$\Rightarrow \mathrm{a}=\frac{30}{5}$
So we get,
$\mathrm{a}=6$
Now we have taken the third term as $\mathrm{a}$ .

Therefore, the third term is $\mathrm{a}=6$.