Question

The first term of an AP is 5, last term is 45 and the sum is 400. Find the number of terms and the common difference.
(a) n = 16 and \[d=\dfrac{8}{3}\].
(b) n = 16 and \[d=\dfrac{16}{3}\].
(c) n = 8 and \[d=\dfrac{16}{3}\].
(d) n = 8 and \[d=\dfrac{8}{3}\].

Answer 1

Hint: Given a and l. Using the sum of n terms formula find the value of n, which is the number of n terms. Similarly, using the sum of n terms formula connecting common differences, find the value of d.


Complete step-by-step answer:
Given that the first term of an AP is 5. The first term is represented as ‘a’. Therefore, a = 5.
The last term of an AP is 45. The last term is represented as ‘l’.
Therefore, l = 45.
In this question we need to find the number of terms, denoted as ‘n’ and the common difference ‘d’.
As the first and last term of the AP is given, we can use the below formula to find the sum of n terms.
Hence, sum of n terms, \[{{S}_{n}}=\dfrac{n}{2}\left( a+l \right)\].
We have been given the sum of n terms, \[{{S}_{n}}=400\].
\[\therefore {{S}_{n}}=\dfrac{n}{2}\left( a+l \right)\]
We know the value of \[{{S}_{n}},a\]and l. So find the value of ‘n’.
Hence, find a number of terms.
\[{{S}_{n}}=\dfrac{n}{2}\left( a+l \right)\]
\[\begin{align}
  & 400=\dfrac{n}{2}\left( 5+45 \right) \\
 & 400=\dfrac{50n}{2} \\
\end{align}\]
\[\therefore \]Cross multiply and simplify the above expression,
\[n=\dfrac{400\times 2}{50}=16\]
Hence, the number of terms, n = 16.
Now we need to find the common difference ‘d’.
So we can use the formula,
Sum of n – terms, \[{{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\].
We know, \[{{S}_{n}}=400\], n = 16, a = 5.
\[\begin{align}
  & \therefore 400=\dfrac{16}{2}\left[ 2\times 5+\left( 16-1 \right)d \right] \\
 & 400=8\left[ 10+15d \right] \\
\end{align}\]
Let us simplify the above expression,
\[\begin{align}
  & \dfrac{400}{8}=10+15d \\
 & 10+15d=50 \\
 & 15d=50-10 \\
 & 15d=40 \\
 & d=\dfrac{40}{15}=\dfrac{8}{3} \\
\end{align}\]
Hence we get a common difference, \[d=\dfrac{8}{3}\].
Thus the number of terms in AP = 16 and common difference, \[d=\dfrac{8}{3}\].
\[\therefore \] Option (a) is the correct answer.

Note: It’s given in the question that we have an AP. So don’t confuse the formulae between an AP and GP. As the last term is given, use the formula connecting it to find the value of n. Similarly, to get the value of d, use the formula connecting to common differences.