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

Answer

Verified

360.9k+ views

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 ${{a}_{n}}=a+(n-1)d$ .

Complete step-by-step answer:

To find the 25

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 25

$\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}$

Hence, the answer is 370.

Note: For solving questions of an A.P. there are two variables first term, common difference and nth term. We need all of these to solve any problem of an A.P and for this we need three equations for these three variables. For this question two variables i.e. first term and n were already given to us and one equation was also given to find common differences.

Complete step-by-step answer:

To find the 25

^{th}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 25

^{th}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}$

Hence, the answer is 370.

Note: For solving questions of an A.P. there are two variables first term, common difference and nth term. We need all of these to solve any problem of an A.P and for this we need three equations for these three variables. For this question two variables i.e. first term and n were already given to us and one equation was also given to find common differences.

Last updated date: 22nd Sep 2023

•

Total views: 360.9k

•

Views today: 9.60k

Recently Updated Pages

What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE