nth Term of AP

Arithmetic progression is a sequence of a pattern of numbers where every term of the sequence has a common difference between them, That difference is always constant for all terms.

The n^th term of an A.P. is called its general term and is represented by a_n.
The formula for the n^th term of A.P. is :

Note: This formula allows you to calculate any term in the sequence if you know the first term and the common difference.

N^th term of an A.P. from the End

To find the n^th term from the end of the A.P., Consider an A.P. where:

• The first term is a,
• The common difference is d,
• The A.P. has m terms.

The n-th term from the end is equivalent to the (m − n + 1)^th term from the beginning. Using the formula for the k-th term of an A.P. (a_k = a + (k − 1)d, the n-th term from the end can be written as:

 a_m-n+1 = a + (m – n)d 

Solved Questions on the n^th Term of AP

Question 1: Check whether progression 11, 10,  9,  8,  5....  is an A.P. or not.
Solution:

The  given progression is 11, 10, 9, 8, 5....

Now, we have to check that d is constant for all term or not.. if it's vary then it is not an A.P. 
d = a₂ - a₁ =  10- 11 = -1   and  d = a₅ - a₄ = 5 - 8 = -3

Common difference is not same for all .so, it's not an A.P. 

Question 2: Which term of the A.P. 11, 17,  23, ........is 551.
Solution:

Here, a = 11 , d = 17 - 11 =  6 

a_n = 551
a_n = a + (n - 1)d = 551 
11 + (n - 1)6 = 551
11 + 6n - 6 = 551 
6n = 546 
n = 91 

Question 3: Is 50 a term of sequence 3, 7, 11, .........
Solution: 

Let us assume 50 is nth term of this sequence or an A.P.
a_n = 50 

Given, a = 3 , d = 4 
a_n = a + (n - 1) d
50 = 3 + (n-1)4 
50 = 3 + 4n - 4 
51 = 4n 
12.7 = n 

Since n must be a natural number (whole number), 12.7 is not valid.

Hence, 50 is not a term of this sequence.

Question 4: Determine the number of terms in the progression 3, 7, 11, ......., 407. Also, find its 10th term from the end. 
Solution: 

A.P.: 3 , 7 , 11 , ........., 407

Given, a = 3 , d= 4  and a_n = 407

an = a+ (n - 1)d 
407 = 3 + (n - 1)4
407 = 3 + 4n - 4
408 = 4n 
102 = n 

Now, it's 10th term form the end is a_{102-10 + 1} = a₉₃

10th term from the end = 3 + 92(3) = 3 + 276 = 371

Question 5: For the A.P. n-1, n- 2, n - 3, ........, find a_m.
Solution:  

A.P: n-1, n-2, n-3, ..........

Here, a = n-1 , d  = (n-2) - (n-1) = -1
a_m = a + (m - 1)d
a_m = n-1 +(m - 1)(-1) 
a_m = n-1 - m + 1 

a_{m =} n - m

Question 6: Which term in the A.P. 5, 2, -1, .........is - 22? 
Solution:

A.P.: 5, 2, -1 ,...........

Given, a = 5 , d = -3 
Now, we have to check - 22 is it's term or not . 
a_n = a + (n - 1)d 
a_n = 5 + (n - 1)(-3)
-22 = 8 -3n
30 = 3n
n = 10

The 10th term of the A.P. 5, 2, −1,…5, 2, -1, is −22.

Question 7: Show that the sequence 7, 2, -3,.......... is an A.P.  Find the general term.
Solution: 

Let sequence 7, 2, -3 ........
if it's an A.P. then d is constant for all

d = a₃ - a₂ = -3 -2 = -5 
d = a₂ - a₁ = 2-7 = -5 

Hence, it is an A.P 

General term of an A.P. is a_n 
a_n = a + (n - 1)d 
a_n = 7 + (n - 1) (-5) 
a_n = 7 + -5n + 5 

 a_{n =} 12- 5n

Read More,

• Arithmetic Sequence
• Arithmetic Series
• Common Difference of AP
• Practice Questions on Arithmetic Sequence