What is the nth term?

Before Arithmetic progression we have to know about Sequences and Series. A sequence is basically a pattern or we can say arrangement of numbers with a certain difference in between them, Not only difference there may be a definite law or formula to find that element. i.e.  1 , 3 , 5 , 7 ….. In this sequence, you will see a  certain pattern or you can say a formula to get these sequences. Its standard formula would be 2n-1. here, n∈ Natural no. . if we put a value on n then we get our sequence. Now come to Series, Series is nothing but just an expression of sequences. Sequence terms connected by + or – sign to form a series. i.e. 1+ 3+ 5 +7 +…..

Arithmetic Progression  

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. This constant difference is denoted by d. 

let us assume an A.P a₁, a₂, a₃, a₄ …….

The general representation of an A.P is a, a+d, a+2d, a+3d,…………

here a = a₁ = first term of A.P and d = common difference. which is easily calculated by subtracting any term with its preceding term. i.e. d = a₂ – a₁ or a₄ – a₃

General Term Of An A.P(nth term of an A.P)

The nth term of an A.P. is called its general term. Nth term may be of a natural number, nth term is the last term of any sequence or arithmetic progression. Nth term of A.P. is calculated by the formula

a_n = a + (n – 1)d

here,

a_n = nth  term of an A.P. 

a = first term 

n = No. of term 

d = common difference. 

Nth term of an A.P. from the end

We can also find the nth term from the end of the A.P., Let us assume an A.P. where the first term is a and d is a common difference having m terms in it. Thus, the nth term from the end is (m-n+1)th term from the beginning. So, we can say a_m-n+1 = a + (m – n)d  is the nth term from the last. 

Sample Problems

Problem 1: Check whether progression 11, 10,  9,  8,  5….  is an A.P. or not. If A.P. find the general term

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. 

Problem 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

Problem 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 

here, a = 3 , d = 4 

a_n = a+ (n-1) d

50 = 3 + (n-1)4 

50 = 3 + 4n – 4 

51 = 4n 

12.7 = n 

n must be an natural number. Because a term is finite number.

Hence. 50 is not a term of this sequence.

Problem 4: Determine the number of terms in the progresssion3, 7, 11, …….,407. Also, find its 10th term from the end. 

Solution: 

A.P.: 3 , 7 , 11 , ………, 407

here, a = 3 , d= 4  and an = 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 = 279

Problem 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} 

Problem 6: Which term in the A.P. 5, 2, -1, ………is – 22? 

Solution:

A.P. 5, 2, -1 ,………..

here, 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 

40 = 3n 

n is not a finite no. so, it’ s not a term of this A.P.

Problem 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