Question

If the first term of an A.P. be $10$, the last term is $50$ and the sum of all the terms is $300$, then the number of terms is
A. 5
B. 8
C. 10
D. 15

Answer 1

Hint: From the given series of arithmetic sequences, we find the general term of the series. We find the formula for ${{t}_{n}}$, the ${{n}^{th}}$ term of the series. We express the sum in the form of first and last term. Then we put the value in the formula to find the solution.

Complete step-by-step answer:
We have been given a series of arithmetic sequences whose first term is 10, the last is 50 and the sum of all the terms is 300. Let there be $n$ terms.
We express the arithmetic sequence in its general form.
We express the terms as ${{t}_{n}}$, the ${{n}^{th}}$ term of the series. The formula being ${{t}_{n}}={{t}_{1}}+\left( n-1 \right)d$.
The first term be ${{t}_{1}}$ and the common difference be $d$ where $d={{t}_{2}}-{{t}_{1}}={{t}_{3}}-{{t}_{2}}={{t}_{4}}-{{t}_{3}}$.
We can express the general term ${{t}_{n}}$ based on the first term and the common difference.
Last term be ${{t}_{l}}$. The general formula for n terms is \[{{S}_{n}}=\dfrac{n}{2}\left[ {{t}_{1}}+{{t}_{l}} \right]\].
So, ${{t}_{1}}=10$, ${{t}_{l}}=50$ and \[{{S}_{n}}=300\].
Putting the values, we get
\[\begin{align}
  & 300=\dfrac{n}{2}\left[ 10+50 \right] \\
 & \Rightarrow n=\dfrac{300\times 2}{60}=10 \\
\end{align}\]
Therefore, the correct option is C.
So, the correct answer is “Option C”.

Note: The sequence is an increasing sequence where the common difference is a positive number. The common difference will never be calculated according to the difference of greater number from the lesser number.