FIND THE NTH TERM OF THE ARITHMETIC SEQUENCE

a) Find the common difference of each arithmetic sequence.

b)  Write the n^th term of each sequence for the given value of n.

Problem 1 :

3, 6, 9, 12, …, n = 8

Solution :

a) Common difference :

a₁ = 3, a₂ = 6, a₃ = 9

d = a₂ – a₁ = 6 - 3 = 3

So, the common difference is 3.

b) Finding the 8^th term :

a_n = a₁ + (n – 1)d

a₈ = 3 + (8 – 1)3

= 3 + 7(3)

= 3 + 21

a₈ = 24

So, the 8^th term of the sequence is a₈ = 24.

Problem 2 :

2, 7, 12, 17, …, n = 12

Solution :

a) Common difference :

a₁ = 2, a₂ = 7, a₃ = 12

d = a₂ – a₁ = 7 - 2 = 5

So, the common difference is 5.

b) n^th term of sequence :

a_n = a₁ + (n – 1)d

Finding 12th term :

a₁₂ = 2 + (12 – 1)5

= 2 + 11(5)

= 2 + 55

a₁₂ = 57

So, the nth term of the sequence is a₁₂ = 57.

Problem 3 :

18, 16, 14, 12, …, n = 10

Solution :

a) Common difference :

a₁ = 18, a₂ = 16, a₃ = 14

d = a₂ – a₁ = 16 - 18 = -2

So, the common difference is -2.

b) 10^th term of sequence :

a_n = a₁ + (n – 1)d

a₁₀ = 18 + (10 – 1)(-2)

= 18 + 9(-2)

= 18 - 18

a₁₀ = 0

So, the 10^th term of the sequence is a₁₀ = 0. 

Problem 4 :

1/2, 1, 3/2, 2, …, n = 7

Solution :

a) Common difference :

a₁ = 1/2, a₂ = 1, a₃ = 3/2

d = a₂ – a₁ = 1 – 1/2

= (2 – 1)/2

 = 1/2

So, the common difference is 1/2.

b) 7^th term of sequence :

a_n = a₁ + (n – 1)d

a₇ = 1/2 + (7 – 1)(1/2)

= 1/2 + 6(1/2)

= 1/2 + 6/2

a₇ = 7/2

So, the nth term of the sequence is a₇ = 7/2.

Problem 5 :

-1, -3, -5, -7, …, n = 10

Solution :

a) Common difference :

a₁ = -1, a₂ = -3, a₃ = -5

d = a₂ – a₁ = -3 + 1 = -2

So, the common difference is -2.

b) 10^th term of sequence :

a_n = a₁ + (n – 1)d

a₁₀ = -1 + (10 – 1)(-2)

= -1 + 9(-2)

= -1 - 18

a₁₀ = -19

So, the nth term of the sequence is a₁₀ = -19.

Problem 6 :

2.1, 2.2, 2.3, 2.4, …, n = 20

Solution :

a) Common difference :

a₁ = 2.1, a₂ = 2.2, a₃ = 2.3

d = a₂ – a₁ = 2.2 – 2.1 = 0.1

So, the common difference is 0.1.

b) 20^th term of sequence :

a_n = a₁ + (n – 1)d

a₂₀ = 2.1 + (20 – 1)(0.1)

= 2.1 + 19(0.1)

= 2.1 + 1.9

a₂₀ = 4

So, the nth term of the sequence is a₂₀ = 4.