Question

In an AP,
Given, $a = 5,d = 3,{a_n} = 50$ , find $n$ and ${S_n}$.

Answer 1

Hint: Here in this question we need to find $n$ and ${S_n}$, and for this ${a_n} = a + \left( {n - 1} \right)d$ and also to find the ${S_n}$, we will use the formula ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - a} \right)d} \right)$. And by using both these formulas and solving it, we can easily solve such questions.

Formula used:
${n^{th}}$ term of an A.P
${a_n} = a + \left( {n - 1} \right)d$
Sum of $n$ terms of an A.P
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - a} \right)d} \right)$
Here,
${a_n}$, will be the ${n^{th}}$ term of an A.P
\[{S_n}\], will be the sum of ${n^{th}}$ term of an A.P
$a$, will be the first term of an A.P
$n$, will be the number of terms
$d$, will be the common difference

Complete step by step answer:
Here in this question we have the values, as $a = 5,d = 3,{a_n} = 50$.
Therefore, by using the formula of ${a_n}$, we will get the equation as
$ \Rightarrow 50 = 5 + \left( {n - 1} \right) \times 3$
Now on solving, the RHS we will get the equation as
$ \Rightarrow 50 = 5 + 3n - 3$
Now on solving it, we get
$ \Rightarrow 50 = 2 + 3n$
Solving furthermore, we will get
$ \Rightarrow 48 = 3n$
And therefore,
$ \Rightarrow n = \dfrac{{48}}{3}$
And on solving it, we get
$ \Rightarrow n = 16$
Therefore, the value of $n$ will be equal to $16$ .
Now we will find the value for \[{S_n}\]
So substituting the value in the formula of \[{S_n}\], we will get the equation as
$ \Rightarrow {S_n} = \dfrac{{16}}{2}\left( {2 \times 5 + \left( {16 - 1} \right)3} \right)$
Now on solving it, we get
$ \Rightarrow {S_n} = 8\left( {2 \times 5 + \left( {15} \right)3} \right)$
Solving furthermore, we will get the equation as
$ \Rightarrow {S_n} = 8 \times 55$
On multiplying it, we get
$ \Rightarrow {S_n} = 440$

Therefore, the value of ${S_n}$ will be equal to $440$.

Note:
Here in this question we had used the term A.P. It stands for arithmetic progression. It is defined as a sequence of numbers in which each of the numbers has the common difference by a constant value.