Question

If the first term of a finite A.P. is $5$, the last term is $45$ and the sum is $500$. Find the number of terms. If the first term and last term of a finite A.P. are $5$ and $95$ respectively and $d = 5$, find $n$ and ${S_n}$.

Answer 1

Use sum of the first term and last term of an AP then substitute the value of first term, last term and sum in the formula of sum of the first term and last term of an AP then find the value of $n$. Again, use the sum of the first $n$ terms of an AP and find the value of $n$ and ${S_n}$.

Complete step by step answer:
Given, The first term of a finite A.P. $a$ is 5.
The last term of a finite A.P. $l$ is 45.
The sum of a finite A.P. ${S_n}$ is 500.
We have, sum of first n terms of an A.P. is given as, ${S_n} = \dfrac{n}{2}\left( {a + l} \right)$ …(i)
Where, Sn is sum of first n terms, n is number of terms, a is first term and l is the last term.
Substitute the value of $a = 5$, ${S_n} = 500$ and $l = 5$ in equation (i), we have

$$500 = \dfrac{n}{2}\left( {5 + 45} \right) \\\ \Rightarrow 1000 = 50n \\\$$

Divide by $50$ on both the sides.
$\dfrac{{50n}}{{50}} = \dfrac{{1000}}{{50}} \Rightarrow n = \dfrac{{100}}{5} \Rightarrow n = 20$
Therefore, the number of terms of a finite A.P., n is $20$.
The first term of a finite A.P. $a$ is 5.
The last term of a finite A.P. $l$ is 95.
The common difference of a finite A.P. $d$ is 5.
The sum of the first $n$ terms of an AP is given by ${S_n} = \dfrac{n}{2}\left( {a + l} \right)$.
Substitute the value of $a = 5$ and $l = 95$ in ${S_n} = \dfrac{n}{2}\left( {a + l} \right)$.
${S_n} = \dfrac{n}{2}\left( {5 + 95} \right) = \dfrac{n}{2} \cdot 100 = 50n$
So,
$\dfrac{n}{2}\left( {a + \left( {n - 1} \right)d} \right) = 50n$ … (ii)
Substitute the value of $a = 5$ and $d = 5$ in equation (ii).

$$\dfrac{n}{2}\left( {5 + \left( {n - 1} \right)5} \right) = 50n \\\ \Rightarrow \dfrac{1}{2}\left( {5 + 5n - 5} \right) = 50 \\\ \Rightarrow \dfrac{1}{2}\left( {5n} \right) = 50 \\\ \Rightarrow 5n = 100 \\\$$

Divide by 5 on both the sides, we get
$\dfrac{{5n}}{5} = \dfrac{{100}}{5} \Rightarrow n = 20$
Therefore, the number of terms of a finite A.P. is $20$.
Substitute the value of $n = 20$ in ${S_n} = 50n$.
${S_n} = 50n = 50 \times 20 = 1000$

Therefore, the sum of first $20$ terms of an AP is $1000$.

Note:
In these types of questions, use formulas of AP very carefully. First see, what elements are given in question, and then choose the appropriate formula, because one formula can give value of only one variable.