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Question: If the sum of a certain number of terms of the A.P. 25, 22, 19... is 116. Find the last term....

If the sum of a certain number of terms of the A.P. 25, 22, 19... is 116. Find the last term.

Explanation

Solution

Here we go through by the properties of arithmetic progression (AP). Here we apply the formula of summation of n terms of AP i.e. Sn=n2(2a+(n−1)d) {S_n} = \dfrac{n}{2}(2a + (n - 1)d){\text{ }}where the sum is given and the term is also given. To find the numbers of terms. And after that we will find any term by the general formula of AP. I.e. Tn=a+(n−1)d{T_n} = a + (n - 1)d

Complete step-by-step answer :
Here in the question it is given that the sum of a certain number of terms of the A.P. 25, 22, 19... is 116.
By the series we can clearly see that the first term is 25 and the second term is 22.
Now we have to calculate the common difference (d),
As we know that we can calculated by subtracting the first term from second term i.e. d=22-25=-3.
So we can write that from question,
Sn=116{S_n} = 116 (Sum of all the terms)
a=25
d=-3
Now we will apply the formula of summation of n terms of AP.
Sn=n2(2a+(n−1)d) {S_n} = \dfrac{n}{2}(2a + (n - 1)d){\text{ }}
Now by putting the values in the formula we get,
⇒116=n2(2×25+(n−1)(−3))  ⇒116×2=n(50−3(n−1)) ⇒232=n(53−3n) ⇒3n2−53n+232=0 ⇒3n2−24n−29n+232=0 ⇒(3n−29)(n−8)=0 ∴n=293,8  \Rightarrow 116 = \dfrac{n}{2}\left( {2 \times 25 + (n - 1)( - 3)} \right){\text{ }} \\\ \Rightarrow {\text{116}} \times 2 = n\left( {50 - 3(n - 1)} \right) \\\ \Rightarrow 232 = n(53 - 3n) \\\ \Rightarrow 3{n^2} - 53n + 232 = 0 \\\ \Rightarrow 3{n^2} - 24n - 29n + 232 = 0 \\\ \Rightarrow (3n - 29)(n - 8) = 0 \\\ \therefore n = \dfrac{{29}}{3},8 \\\
But n=293n = \dfrac{{29}}{3} can’t be possible because the number of terms can’t be in fraction.
Therefore the number of terms is 8.
Now we have to find out the last term. So we apply the general term of AP. i.e. Tn=a+(n−1)d{T_n} = a + (n - 1)d
⇒T8=25+(8−1)(−3) ⇒T8=25−21=4  \Rightarrow {T_8} = 25 + (8 - 1)( - 3) \\\ \Rightarrow {T_8} = 25 - 21 = 4 \\\
Hence the last term of the given series is 4.

Note : Whenever we face such a type of question the key concept for solving the question is first of all find the total number of terms and then apply the general formula of AP to find that term which it asked. And for finding the total number of terms if the last term and the first term are given we will also apply Sn=n2(a+l) {S_n} = \dfrac{n}{2}(a + l){\text{ }}.