Question

Find the middle term of the given A.P i.e. $213,205,197,.......,37.$

Answer 1

Hint: In order to solve this question, first we have to write the first term and then find the common difference from the given AP, then we will take the help of the last term to find the number of terms and then its middle term by using the concept and formulas of arithmetic progression.

Complete step-by-step answer:
In order to find the middle term of the sequence, first we have to know how many terms are in the given sequence.
In this given AP $213,205,197,.......,37.$
First term $a = 213$
Common difference $d = 205 - 213 = \left( { - 8} \right)$
Last term ${a_n} = 37$
Given, a general formula for the $n_{th}$ term in an AP in terms of the first term, common difference and last term
${a_n} = a{\text{ + }}\left( {n - 1} \right)d$
Now, the number of terms will be
$
  37 = 213 + \left( {n - 1} \right)\left( { - 8} \right) \\
   \Rightarrow 37 = 213 - 8n + 8 \\
   \Rightarrow 8n = 221 - 37 \\
   \Rightarrow n = \dfrac{{184}}{8} \\
  n = 23 \\
 $
Hence, this AP contains 23 terms.
As you know, the middle term for odd numbers can find by using $\dfrac{{n + 1}}{2}$
So the middle term will be $\dfrac{{23 + 1}}{2} = \dfrac{{24}}{2} = 12$
Hence, the middle term of this AP is its $12_{th}$ term.
So, we can find the $12_{th}$ term by using formula
${a_n} = a{\text{ + }}\left( {n - 1} \right)d$ , where $n = 12,{\text{ }}a = 213,{\text{ }}d = - 8$
$
  {a_{12}} = 213 + \left( {12 - 1} \right)\left( { - 8} \right) \\
  {\text{ }} = 213 - \left( {11 \times 8} \right) \\
  {\text{ }} = 213 - 88 \\
  {\text{ }} = 125 \\
 $
So, the middle term of this AP is $12_{th}$ term and the value of $12_{th}$ term is $125$.

Note: First, we should know what the arithmetic sequence or arithmetic progression is which is a sequence of numbers such that the difference between the consecutive terms is constant. A student must put the values carefully in the formula and also has to remember the basic terminology of arithmetic progression. Above mentioned is the best way to get our answer.