Middle Term of an Arithmetic Progression

If we talk about the meaning and introduction of an arithmetic progression, ‘Arithmetic’ means mathematical numbers. On the other hand, ‘Progression’ is referred to as moving from one number to another.

Let us take an example to clear this: 3, 5, 7, 9, 11, 13 are said to be mathematical numbers. If we are moving from one number to another (progression), we have to add ‘2’ to get to the other number.

For instance: 

3+2=5

5+2=7

7+2=9

9+2=11

11+2=13

2 (the number that we are adding) is a constant number. We cannot add 1 or 3 to get this series. Adding a constant number is compulsory. So, this series in which we add a particular number and move from one number to another is called Arithmetic Progression.

Similarly, suppose we have a series of 3, 5, 7, 10, 14. Then, it is not an arithmetic progression. Because

3+2=5, 5+2=7, 7+3=10, 10+4=14

We are adding different numbers to get a series. Moreover, we should only add a constant number to get a series. So, it will not be called an Arithmetic Progression.

Middle term of an Arithmetic Progression 

Above, we have discussed the meaning and introduction of Arithmetic Progression. Now, before discussing how to find the middle term of an Arithmetic Progression, let us see some of the basic formulas:

Formula to Find the nth Term of an A.P. 

an = a + (n-1) × d

an is the nᵗʰ term in the sequence. Moreover, the first term in the sequence

is ‘a. In addition to this, ‘d’ is the common difference between terms.

Sum of n terms in an A.P.

Sn = n/2 [2a + (n-1) × d]

Sum of all terms of an A.P. 

S all terms = n/2 ( a+l)

Middle term of an Arithmetic Progression with ‘n’ terms

If ‘n’ is odd, then the middle term= (n+1)/2 th term

For instance, 15=odd=n

(15+1)/2 th term

16/2 th term

8th term

So, the answer is the 8th term will be the middle term of an Arithmetic Progression.

If ‘n’ is even, then the two middle terms= n/2 th term and n/2 + 1 th term.

For instance, 16=even=n

16/2 th term

8th term and,

16/2 + 1 th term

8+1 th term

9th term

So, the 8th and 9th terms will be a middle term of an Arithmetic Progression.

Find the Middle Term of an Arithmetic Progression

Let us discuss some questions about finding the middle term of an Arithmetic Progression.

Q1) Find the middle term of an Arithmetic Progression 6, 13, 20,……, 216.

Here, 

a = 6 (a is the first term of the sequence)

d = (common difference between terms)=

a² – a¹ = 13-6 = 7

a(n) = 216

a + (n-1)d = 216

So, according to the above-given details,

6+(n-1)7=216

6+7n-7=216

7n-1=216

7n=216+1

7n=217

n=217/7

n=31=odd

So, the value of ‘n’ will be 31= odd

Therefore, Middle term = (n+1)/2th

(31+1)/2th

32/2th

=16th term

a16= a+15d

6+15*7

6+105

111 

So, 111 will be the answer to the above question.

Q2) Find the middle term of an Arithmetic Progression 3,8,13,18,….., 73?

According to the question,

a= 3 (first term of the question)

l= 73 (last term of the question)

d= a² – a¹ (difference between the both)

  =8-3

  =5

l= a+(n-1)d

73= 3+(n-1).5

73-3= (n-1).5

70= 5n-5

70+5=5n

75=5n

n=75/5

n=15

Therefore, the total no. of terms is 15 (odd).

According to the formula:

If ‘n’ is odd, then the middle term= (n+1)/2 th term

(15+1)/2 th term

16/2 th term

8th term

a+(8-1)d

a+7d

3+7*5

3+35

38

So, after following the question, we’ll get 38 as our answer.

Conclusion

In this article, we have the introduction and meaning of A.P., the middle term of A.P., and how to find the middle term of A.P. Moreover, these notes also discuss some examples to clarify the topic. 

Arithmetic Progression is not a difficult topic. To solve the questions, one has to be clear with the concept and formulas. Once you understand the concept, you will be able to score well in the exams.