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Sum of the first 14 terms of an A.P is 1505 and its first term is 10. Find its 25th term.

Last updated date: 30th Mar 2023
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Hint: To find the 25th term we need the first term and the common difference from the given information. We can find the nth term of an A.P. by using the following formula ${{a}_{n}}=a+(n-1)d$ .

To find the 25th term, we have n=25 and we need to find a and d i.e. first term and common difference of the A.P.
The given information is the sum of the first 14 terms of an A.P.
The sum of first n terms of an A.P. is given by ${{S}_{n}}=\dfrac{n}{2}\{2a+(n-1)d\}$ ……… (i) Where again a=first term and d=common difference of the A.P.
We substitute the value of n=14 and a=10 in equation (i)
${{S}_{14}}=\dfrac{14}{2}\{2.10+(14-1)d\}$

From the question we have, ${{S}_{14}}=1505$ , therefore by substituting the value of ${{S}_{14}}$ in the above equation we have,
$1505=\dfrac{14}{2}\{2.10+(14-1)d\}$ ……………. (ii)
From the above equation we can find the value of d
Now we simplify equation (ii)
$1505=7(20+13d)$
Dividing LHS and RHS by 7 we have,
\begin{align} & \dfrac{1505}{7}=20+13d \\ & \Rightarrow 215=20+13d \\ \end{align}

Subtracting 20 both sides we have,
\begin{align} & \Rightarrow 195=13d \\ & \Rightarrow d=\dfrac{195}{13} \\ & \Rightarrow d=15 \\ \end{align}

Therefore, now we have the value of d is equal to 15.
Now we can find the 25th term as we have n, a and d for the formula ${{a}_{n}}=a+(n-1)d$ .
a=10, d=15 and n=25
The 25th term is given by,
\begin{align} & {{a}_{25}}=a+(n-1)d \\ & =10+(25-1)15 \\ \end{align}

On further simplification we have,
\begin{align} & {{a}_{25}}=10+24\times 15 \\ & =10+360 \\ & =370 \\ \end{align}