HOW TO FIND THE SPECIFIED TERM OF AN ARITHMETIC SEQUENCE

Find the n^th term of the sequence, then find the 20^th term.

Problem 1 :

a₁ = 2 and d = 3

Solution :

a₁ = 2 and d = 3

To find the n^th term of the sequence :

a_n = a₁ + (n – 1)d

a_n = 2 + (n – 1)(3)

a_n = 2 + 3n – 3

a_n = 3n – 1

To find the 20^th term of the sequence :

a_n = 3n – 1

a₂₀ = 2 + (20 – 1)3

a₂₀ = 2 + (19)3

= 2 + 57

a₂₀ = 59

So, the 20^th term is a₂₀ = 59.

Problem 2 :

-6, -4, -2, …

Solution :

-6, -4, -2, …

a_n = a₁ + (n – 1)d

d = a₂ – a₁ = -4 + 6 = 2

 = a₃ – a₂ = -2 + 4 = 2

To find the n^th term of the sequence :

a_n = -6 + (n – 1)2

= -6 + 2n – 2

a_n = 2n – 8

To find the 20^th term of the sequence :

a₂₀ = 2(20) – 8

= 40 – 8

a₂₀ = 32

So, the 20^th term is a₂₀ = 32.

Problem 3 :

a₁ = 0 and d = 2/3

Solution :

a₁ = 0 and d = 2/3

To find the n^th term of the sequence :

a_n = a₁ + (n – 1)d

a_n = 0 + (n – 1)(2/3)

a_n = (2/3)(n – 1)

To find the 20^th term of the sequence :

a₂₀ = 0 + (20 – 1)2/3

a₂₀ = (19)2/3

a₂₀ = 38/3

So, the 20^th term is a₂₀ = 38/3.

Problem 4 :

2/5, 1/15, -4/15, …

Solution :

2/5, 1/15, -4/15, …

a_n = a₁ + (n – 1)d

d = a₂ – a₁ = 1/15 – 2/5

= 1/15 – 2/5 × (3/3)

= 1/15 – 6/15

= -5/15

= -1/3

To find the n^th term of the sequence :

a_n = 2/5 + (n – 1)(-1/3)

To find the 20^th term of the sequence :

a_n = 2/5 + (n – 1)(-1/3)

a₂₀ = 2/5 + (20 – 1)(-1/3)

a₂₀ = 2/5 + 19(-1/3)

a₂₀ = 2/5 + (-19/3) (5/5)

= (6/15) - (95/15)

= (6 - 95)/15

a₂₀ = -89/15

So, the 20^th term is a₂₀ = -89/15.