HOW TO FIND NTH TERM OF AN ARITHMETIC SEQUENCE

Examples 1-4 : Find the nth term of the following arithmetic sequences :

Example 1 :

4, 9, 14, …………

Solution :

Common difference :

d = a₂ – a₁

= 9 – 4

= 5

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = 4 and d = 5.

a_n = 4 + (n - 1)(5)

= 4 + 5n - 5

= 5n - 1

Example 2 :

125, 120, 115, 110, …………

Solution :

Common difference :

d = a₂ – a₁

= 120 – 125

= -5

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = 125 and d = -5.

a_n = 125 + (n - 1)(-5)

= 125 - 5n + 5

= 130 - 5n

Example 3 :

24, 23¼, 22½, 21¾, …………

Solution :

Common difference :

d = a₂ – a₁

= 23¼ - 24

= 93/4 - 24

= (93 - 96)/4

= -3/4

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = 24 and d = -3/4.

a_n = 24 + (n - 1)(-3/4)

= 24 - 3n/4 + 3/4

= 24 + 3/4 - 3n/4

= (96 + 3)/4 - 3n/4

= 99/4 - 3n/4

Example 4 :

√2, 3√2, 5√2, …………

Solution :

Common difference :

d = a₂ – a₁

= 3√2 - √2

= 2√2

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = √2 and d = 2√2.

a_n = √2 + (n - 1)(2√2)

= √2 + (2√2)n - 2√2

= (2√2)n - √2

Example 5 :

The 10^th and 18^th terms of an arithmetic sequence are 41 and 73 respectively. Find the n^th term.

Solution :

(2) - (1) :

8d = 32

d = 4

Substitute d = 4 in (1).

a₁ + 9(4) = 41

a₁ + 36 = 41

a₁ = 5

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = 5 and d = 4.

a_n = 5 + (n - 1)(4)

= 5 + 4n - 4

= 4n + 1

Example 6 :

Find n so that the n^th terms of the following two arithmetic sequences are the same.

1, 7, 13, 19, …………

100, 95, 90, …………

Solution :

Part (i) :

1, 7, 13, 19, …………

Common difference :

d = a₂ – a₁

= 7 - 1

= 6

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = 1 and d = 6.

a_n = 1 + (n - 1)(6)

= 1 + 6n - 6

= 6n - 5

Part (ii) :

100, 95, 90, …………

Common difference :

d = a₂ – a₁

= 95 - 100

= -5

n^th term of an arithmetic sequence :

a_n = a₁ + (n - 1)d

Substitute a₁ = 100 and d = -5.

a_n = 100 + (n - 1)(-5)

= 100 - 5n + 5

= 105 - 5n

Solve for n :

Given : n^th terms of the two arithmetic sequences are the same.

6n - 5 = 105 - 5n

11n - 5 = 105

11n = 110

n = 10

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.