Question

If ${n^{th}}$ term of an A.P. is $2n + 1$, what is the sum of its first three terms?

Answer 1

Hint: A.P. is a sequence in which every set of consecutive terms have a common difference. We will first find the first three terms by putting $n = 1,2$ and 3 in the expression of ${n^{th}}$ term of the arithmetic sequence, which is $2n + 1$. Then we will add those terms to find the sum of the first three terms of the A.P.

Complete step-by-step answer:
We are given that the ${n^{th}}$ term of the sequence is $2n + 1$
We have to find the sum of the first three terms.
We will begin by finding the first three terms of the given A.P.
For, first term let $n = 1$, then first term is \[2\left( 1 \right) + 1 = 2 + 1 = 3\]
Now we will substitute $n = 2$ in the given expression to find the second term of A.P.
\[2\left( 2 \right) + 1 = 5\]
Similarly, we will substitute $n = 3$ in the expression to find its third term.
\[2\left( 3 \right) + 1 = 7\]
Hence, the first three terms of the A.P. is 3,5,7
We will add these terms to find the sum of the first three terms of A.P.
\[3 + 5 + 7 = 15\]
Hence, the sum of the first three terms of the A.P is 15.

Note: We can also do this question by finding only first and third term and then apply the formula $\dfrac{n}{2}\left( {{a_1} + {a_n}} \right)$, where ${a_1}$ is the first term, ${a_n}$ is the last term and $n$ is the number of terms. Also, we can find the value of any term using the formula, ${a_n} = {a_1} + \left( {n - 1} \right)d$, where $d$ is the common difference.