Question

Find the middle term of the AP: 213, 205, 197, ………., 37.

Answer 1

Hint: The formula for the n^th term of any AP is given as \[{t_n} = a + (n - 1)d\]. Use this to find the number of terms in the AP. Then find the middle term.

Complete step-by-step answer:

An arithmetic progression (AP) is a sequence of numbers whose consecutive terms differ by a constant number. This constant number is called the common difference of the AP.
We are given the AP as 213, 205, 197, …….., 37. We can find the common difference by subtracting the first term from the second.
\[d = 205 - 213\]
\[d = - 8..........(1)\]
The first term of the AP is 213.
\[a = 213............(2)\]
The formula to calculate the nth term of any AP is given as follows:
\[{t_n} = a + (n - 1)d...........(3)\]
The last term of the AP is 37, we find the total number of terms in the AP using equations (1), (2) and (3).
$\Rightarrow$ \[37 = 213 + (n - 1)( - 8)\]
Simplifying, we have:
$\Rightarrow$ \[37 - 213 = (n - 1)( - 8)\]
$\Rightarrow$ \[ - 176 = (n - 1)( - 8)\]
Dividing both sides by – 8, we have:
$\Rightarrow$ \[\dfrac{{ - 176}}{{ - 8}} = n - 1\]
$\Rightarrow$ \[22 = n - 1\]
Solving for n, we have:
$\Rightarrow$ \[n = 22 + 1\]
$\Rightarrow$ \[n = 23...........(4)\]
The middle term of the AP is found by using equation (4) as follows:
Middle term = \[\dfrac{{n + 1}}{2}\]
Middle term = \[\dfrac{{23 + 1}}{2}\]
Middle term = \[\dfrac{{24}}{2}\]
Middle term = 12
Hence, the 12th term of the AP is the middle term.
Using equation (3), we have:
$\Rightarrow$ \[{t_{12}} = 213 + (12 - 1)( - 8)\]
Simplifying, we have:
$\Rightarrow$ \[{t_{12}} = 213 + (11)( - 8)\]
$\Rightarrow$ \[{t_{12}} = 213 - 88\]
$\Rightarrow$ \[{t_{12}} = 125\]
Hence, the middle term of the AP is 125.

Note: After finding that the number of terms of the AP is an odd number, you can also find the average of the first and the last term which will be equal to the middle term. But note that this is true only for an odd number of terms.