Question

In an arithmetic progression it is given that $a = 2,d = 8,{S_n} = 90$. Find $n$ and ${a_n}$.

Answer 1

Hint: Firstly, we have to know that an A.P (Arithmetic Progression) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant that is called $d$.
Here we will find the value of “$n$” by the sum formula of A.P and then we will get the value of ${a_n}$ term.

Formula used:
${S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$
The \[nth\] term of the arithmetic progression ${a_n} = a + (n - 1)d$

Complete step by step answer:
It is given that, $a = 2,d = 8,{S_n} = 90$
We have to find out the value of $n = ?$ and ${a_n} = ?$
As we know the formula for ${S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$
On substituting the value of${S_n}$ in the given data and we get,
$\Rightarrow 90 = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$
We need to put the values of $a$ and $d$
$\Rightarrow 90 = \dfrac{n}{2}\left[ {2 \times 2 + (n - 1)8} \right]$
On doing cross multiplying and multiply the terms of the brackets and we get,
$\Rightarrow 180 = n(4 + 8n - 8)$
On subtracting the integer, we get
$\Rightarrow 180 = n(8n - 4)$
Now we must open the brackets to multiplying the terms
$\Rightarrow 180 = 8{n^2} - 4n$
We just put all the terms on the left-hand side of zero
$\Rightarrow 8{n^2} - 4n - 180 = 0$
On dividing $4$ we get
$\Rightarrow 2{n^2} - n - 45 = 0$
Now, factorization of the above equation,
We get,
$\Rightarrow 2{n^2} - 10n + 9n - 45 = 0$
Taking the$2n$as common in the first two terms and $9$ as common in the last two terms we get
$\Rightarrow 2n(n - 5) + 9(n - 5) = 0$
On taking the common terms we get,
$\Rightarrow (2n + 9)(n - 5) = 0$
We need to find the values of $n$ by putting both the equations
$\Rightarrow 2n + 9 = 0$ and $n - 5 = 0$
$\Rightarrow 2n = - 9$ and $n = 5$
$\Rightarrow n = \dfrac{{ - 9}}{2}$ and $n = 5$
We cannot take the negative value for $n,$ so we will take value for $n$ is positive.
Hence, we will take the value for $n = 5$
Therefore, the value of $n = 5$
Now we need to find the ${a_n}$
Thus, ${a_n} = a + (n - 1)d$
Putting the value of all the variables and we get,
$\Rightarrow =2 + (5 - 1)8$
On subtracting the bracket terms and multiply it we get
$\Rightarrow = 2 + 32$
On adding the terms we get
$\Rightarrow a_n = 34$

$\therefore$The value of $n$ is 5. $n^{th}$ term of the given A.P. is $a_n=a_5=34$.

Note:
From the given sequence, we can easily read out the first term and common difference $d$
Regular contrast of any pair of back to back or contiguous numbers.
There are three things to need to examine for calculating the $n^{th}$ term by using the formula, the first term $\left( a \right)$ and the common difference between consecutive terms $\left( d \right)$ and the term
The \[nth\] term of the arithmetic sequence is in the form of \[an + b\], then the sequence is in the arithmetic progression and the common difference is $d$.