Question

For a given AP, a = 2, d = 8 and Sñ= 90. Find n and an​

Answer 1

Given,

An A.P in which a=2

d=8

Sn = 90

To Find,

The value of n and aₙ

Solution,

Since the sum of n terms is given

Sₙ = n/2(2a+(n-1)d)

90 = n/2(4+8n-8)

90 = n(4n-2)

90 = 4n²-2n

4n²-2n-90 = 0

After solving the quadratic equation we get n = 5

Now,

aₙ = a⁵ = a+4d

           = 2+32

           = 34

Hence, the value of n is 5 and aₙ is 34.

Answer 2

Given,

For an AP,

a = 2,

d = 8, and,

$S_n=90.$

To find,

n, and $a_n$.

Solution,

We can solve the given numerical problem by the process given below.

So, an arithmetic progression, or AP is defined as a sequence or series of numbers that are in such a way that the difference between any two consecutive numbers is the same or equal to the difference between any other two consecutive numbers. This difference is called the common difference.

In an AP, the following representation is used for various terms,

a = first term,

d = common difference $=[a_{n+1}-a_n]$,

n = number of terms,

$a_n$ = the $n^{th}$ term, or the term at $n^{th}$ the position, given by,

$a_n=a+(n-1)d$, and,

$S_n=$ the sum of n terms, given by,

$S_n=\frac{n}{2} [2a+(n-1)d]$

Here, we can find out n using the formula for $S_n$. Substituting respective values,

$90=\frac{n}{2} [2(2)+(n-1)8]$

$180 = n[4+8n-8]$

$180=n[8n-4]$

Simplifying and rearranging,

$8n^{2} -4n-180=0\\2n^{2} -n-45=0$

On solving the above quadratic equation, we get,

n = 5, -4.5.

Since the number of terms can't be negative or fraction,

so only n = 5 will be considered.

So, the number of terms in the given AP = 5.

Now, for n = 5, we can find out $a_n$ using the aforementioned formula,

$a_n=a+(n-1)d$

Substituting values for a, n, and d, we get,

$a_9=2+(5-1)8$

⇒ $a_9=34$.

Therefore, for the given AP, n = 5, and $a_n\hspace{3} or \hspace{3} a_9=34$.