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Question: In an AP, Given, \(a = 5,d = 3,{a_n} = 50\) , find \(n\) and \({S_n}\)....

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

Explanation

Solution

Here in this question we need to find nn and Sn{S_n}, and for this an=a+(n−1)d{a_n} = a + \left( {n - 1} \right)d and also to find the Sn{S_n}, we will use the formula Sn=n2(2a+(n−a)d){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:
nth{n^{th}} term of an A.P
an=a+(n−1)d{a_n} = a + \left( {n - 1} \right)d
Sum of nn terms of an A.P
Sn=n2(2a+(n−a)d){S_n} = \dfrac{n}{2}\left( {2a + \left( {n - a} \right)d} \right)
Here,
an{a_n}, will be the nth{n^{th}} term of an A.P
Sn{S_n}, will be the sum of nth{n^{th}} term of an A.P
aa, will be the first term of an A.P
nn, will be the number of terms
dd, will be the common difference

Complete step by step answer:
Here in this question we have the values, as a=5,d=3,an=50a = 5,d = 3,{a_n} = 50.
Therefore, by using the formula of an{a_n}, we will get the equation as
⇒50=5+(n−1)×3\Rightarrow 50 = 5 + \left( {n - 1} \right) \times 3
Now on solving, the RHS we will get the equation as
⇒50=5+3n−3\Rightarrow 50 = 5 + 3n - 3
Now on solving it, we get
⇒50=2+3n\Rightarrow 50 = 2 + 3n
Solving furthermore, we will get
⇒48=3n\Rightarrow 48 = 3n
And therefore,
⇒n=483\Rightarrow n = \dfrac{{48}}{3}
And on solving it, we get
⇒n=16\Rightarrow n = 16
Therefore, the value of nn will be equal to 1616 .
Now we will find the value for Sn{S_n}
So substituting the value in the formula of Sn{S_n}, we will get the equation as
⇒Sn=162(2×5+(16−1)3)\Rightarrow {S_n} = \dfrac{{16}}{2}\left( {2 \times 5 + \left( {16 - 1} \right)3} \right)
Now on solving it, we get
⇒Sn=8(2×5+(15)3)\Rightarrow {S_n} = 8\left( {2 \times 5 + \left( {15} \right)3} \right)
Solving furthermore, we will get the equation as
⇒Sn=8×55\Rightarrow {S_n} = 8 \times 55
On multiplying it, we get
⇒Sn=440\Rightarrow {S_n} = 440

Therefore, the value of Sn{S_n} will be equal to 440440.

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.