Question

Find the middle term of the A.P. 6, 13, 20, …… 216.

Answer 1

Before solving this question, we must know about how to represent the ${{n}^{th}}$ term of an arithmetic progression which is as follows
${{a}_{n}}=a+(n-1)d$
(Where a is the first term of the arithmetic progression and d is the common difference of the arithmetic progression)
In this question, we will first find the total number of elements that are present in the given arithmetic progression using the general term formula that is ${{a}_{n}}=a+(n-1)d$ . Then, we can find the middle term of the arithmetic progression easily by first finding the value of n and then we will get the value of the middle term by putting the value of n for the middle term in the general term formula that is ${{a}_{n}}=a+(n-1)d$ .

Complete step by step answer:
As mentioned in the question, we have to find the middle term of the given arithmetic progression.
Now, as mentioned in the hint, we can write the ${{n}^{th}}$ term of an arithmetic progression as follows

& \Rightarrow {{a}_{n}}=a+(n-1)d \\\ & \\\ \end{aligned}$$ We know that, First term, $a=6$ Common difference, $d=7$ ${{n}^{th}}$ term ,${{a}_{n}}=216$ Hence, we can write the equation for ${{n}^{th}}$ term as, $$\begin{aligned} & \Rightarrow 216=6+(n-1)7 \\\ & \Rightarrow 216=6+7n-7 \\\ & \dfrac{\Rightarrow 217}{7}=n \\\ & \Rightarrow n=31 \\\ \end{aligned}$$ Since, n is an odd number, $$middle\ term=\left( \dfrac{n+1}{2} \right)=\dfrac{31+1}{2}={{16}^{th}}\,term$$ Now, we can calculate the value of the middle term of the arithmetic progression by putting the value of n which we got from above as follows $$\Rightarrow {{a}_{middle\,term}}=a+(n-1)d$$ Here, $a=6$,$d=7$, $n=16$ $$\Rightarrow {{a}_{16}}=6+(16-1)7$$ $$\Rightarrow {{a}_{16}}=6+105=111$$ **Hence, the middle term of the given arithmetic progression is 111.** **Note:** The formula for the middle term when the total number of terms are odd differs from the formula for middle term when the total number of terms are even. Total number of terms being ‘n’, when the number of terms are odd, $$middle\ term={{\left( \dfrac{n+1}{2} \right)}^{th}}term$$ When the number of terms are even, we will have two middle terms, that is, $first\,middle\,term={{\left( \dfrac{n}{2} \right)}^{th}}term$ $second\,middle\,term={{\left( \dfrac{n}{2}+1 \right)}^{th}}term$