Question

Find the middle term of the A.P. 213, 205, 197, …, 37.

Answer 1

In order to solve this problem, we need to find constant spacing in the sequence. In arithmetic progression, the ${{n}^{th}}$ term is calculated by finding with the help of formula given as ${{t}_{n}}=a+\left( n-1 \right)d$ where, n is the number of terms, a = first term, d = spacing between two successive numbers.
The formula for the middle term in case the $n$ is odd is given by $\dfrac{n+1}{2}$.
The formula for the middle term in case the $n$ is even is given by the average of $\dfrac{n}{2}$ and $\dfrac{n+1}{2}$ term.

Complete step-by-step solution:
We know that the following numbers are in arithmetic progression.
An arithmetic progression is a sequence of numbers such that the difference of any two numbers successive members is a constant.
We know that the spacing between the two numbers is constant.
Let the distance between two successive numbers be (d).
Therefore, $d = 205 – 213 = -8$.
We can see that first term and we know the last term.
Let the first term be $(a) = 213$.
Let the ${{n}^{th}}$ term be $\left( {{t}_{n}} \right)$ .
The formula for finding the ${{n}^{th}}$ is ${{t}_{n}}=a+\left( n-1 \right)d$
Where n is the number of terms,
a = first term
d = spacing between two successive numbers.
Substituting the values, we get,
$37=213+\left( n-1 \right)\times \left( -8 \right)$
Solving for n, we get,
$\begin{aligned} & 37=213-8n+8 \\\ & 37=221-8n \\\ & 8n=184 \\\ & n=\dfrac{184}{8}=23 \\\ \end{aligned}$
To find the middle term, we have separate formulas when $n$ is even and when $n$ is odd.
The formula for the middle term in case the $n$ is odd is given by $\dfrac{n+1}{2}$ .
The formula for the middle term in case the $n$ is even is given by the average of $\dfrac{n}{2}$ and $\dfrac{n+1}{2}$ term.
As we can see that $n$ is odd, the middle term is $=\dfrac{23+1}{2}={{12}^{th}}$ term.
Now, we need to find the value of the ${{12}^{th}}$ term.
Using the same formula ${{t}_{n}}=a+\left( n-1 \right)d$ , with n = 12, we get,
$\begin{aligned} & {{t}_{12}}=213+\left( 12-1 \right)\times \left( -8 \right) \\\ & =213+11\left( -8 \right) \\\ & =213-88 \\\ & =125 \end{aligned}$
Therefore, the middle term in the given sequence is 125.

Note: We need to find the middle term carefully. It can be calculated by just dividing the number by two and considering the next integer. In this case, the middle number of 23 is $\dfrac{23}{2}=11.5$, now the middle term is 12. Also notice that the sequence is decreasing therefore, the distance between numbers is decreasing.