Question

Find the sum to n terms of the A.P. whose ${{\text{k}}^{th}}$ term is $5{\text{k + 1}}$.

Answer 1

Hint: We will form the series of A.P. using the given term. We will put the values of k as $1,2,3,...$ to get the terms of the series. Then we will use the formula of sum of n terms of A.P. series which is given as-
$ \Rightarrow {{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left[ {2{\text{a + }}\left( {{\text{n}} - {\text{1}}} \right){\text{d}}} \right]$
Here n is the number of the terms, ‘a’ is the first term and d is the common difference between the terms of the series. Solve the formed equation and we’ll get the answer.

Complete step-by-step answer:
Given, the ${{\text{k}}^{th}}$ term of an arithmetic series is $5{\text{k + 1}}$. We have to find the sum of n terms of this series. So first we will find the series. We can write-
$ \Rightarrow {{\text{a}}_{\text{k}}} = 5{\text{k + 1}}$ - (i)
On putting k$ = 1$ we get-
$ \Rightarrow {{\text{a}}_1} = 5{\text{ + 1}}$
On adding the given terms, we get-
$ \Rightarrow {{\text{a}}_1} = 6$
On putting k$ = 2$ we get-
$ \Rightarrow {{\text{a}}_2} = \left( {5 \times 2} \right){\text{ + 1}}$
On solving, we get-
$ \Rightarrow {{\text{a}}_2} = 10 + 1$
On adding the given terms, we get-
$ \Rightarrow {{\text{a}}_2} = 11$
On putting k$ = 3$ we get-
$ \Rightarrow {{\text{a}}_3} = \left( {5 \times 3} \right){\text{ + 1}}$
On solving, we get-
$ \Rightarrow {{\text{a}}_3} = 15 + 1$
On adding the given terms, we get-
$ \Rightarrow {{\text{a}}_3} = 16$
Similarly, on continuing, we get a series whose terms are $6$,$11$,$16$,…
In this series, first term ${\text{a = 6}}$ and common difference ${\text{d = 11 - 6 = 16 - 11 = 5}}$
So this series is in A.P.
Now, we will find the sum of n terms of this series using the formula-
$ \Rightarrow {{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left[ {2{\text{a + }}\left( {{\text{n}} - {\text{1}}} \right){\text{d}}} \right]$
Here n is the number of the terms, ‘a’ is the first term and d is the common difference between the terms of the series.
On putting the given values in the formula, we get-
$\Rightarrow {{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left[ {\left( {2 \times 6} \right){\text{ + }}\left( {{\text{n}} - {\text{1}}} \right)5} \right]$
On solving, we get-
$ \Rightarrow {{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left[ {12{\text{ + }}5{\text{n}} - 5} \right]$
On further solving, we get-
$ \Rightarrow {{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left[ {{\text{7 + }}5{\text{n}}} \right]$
The sum of n terms of the given A.P. is $\dfrac{{\text{n}}}{2}\left[ {{\text{7 + }}5{\text{n}}} \right]$.

Note: We can also solve this question using the formula which gives the relation between the first term and last term of A.P. Here clearly the first term will be $6$ and we can find the last term by putting k=n. Then use the formula of sum of n terms of A.P. which is given as-
$ \Rightarrow {{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left[ {a + l} \right]$
Put the required values in the above formula to get the answer.