Answer

Verified

417.9k+ views

**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 $d$.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One cusec is equal to how many liters class 8 maths CBSE