FIND FIRST FIVE TERMS OF SEQUENCE WHEN nth TERM IS GIVEN

Write the first five terms of the sequence. Determine whether or not the sequence is arithmetic. If it is, find the common difference.

Problem 1 :

a_n = 8 +13n

Solution :

a_n = 8 +13n

a₅ = 8 +13(5)

= 8 + 65

a₅ = 73

So, first five terms of the sequence is 21, 34, 47, 60, 73.

To find the common difference :

d = a₂ – a₁ = 34 – 21 = 13

a₃ – a₂ = 47 – 34 = 13

The sequence is arithmetic because common difference is 13.

Problem 2 :

a_n = (1/n) +1

Solution :

a_n = (1/n) +1

a₅ = (1/5) +1)

a₅ = 6/5

So, first five terms of the sequence is 2, 3/2, 4/3, 5/4, 6/5.

To find the common difference :

d = a₂ – a₁ = (3/2) - 2

= (3 - 4)/2

d = -1/2

Problem 3 :

a_n = 2ⁿ + n

Solution :

a_n = 2ⁿ + n

So, first five terms of the sequence is 3, 6, 11, 20, 37.

To find the common difference :

d = a₂ – a₁ = 6 – 3 = 3

a₃ – a₂ = 11 – 6 = 5

Since the common difference is not same, it is not arithmetic sequence.

Find the first four terms of the sequence given the explicit formula.

Problem 1 :

a_n = n² + 3

Solution :

a_n = n² + 3

a₄ = 4² + 3

= 16 + 3

a₄ = 17

So, first four terms of the sequence is 4, 7, 12, 17.

Problem 2 :

a_n = 15/(n + 3)

Solution :

a_n = 15/(n + 3)

a₄ = 15/(4 + 3)

a₄ = 15/7

a₄ = 15/7

So, first four terms of the sequence is 15/4, 3, 5/2, 15/7.

Problem 3 :

a_n = 2n - 1

Solution :

a_n = 2n - 1

a₄ = 2(4) – 1

= 8 – 1

a₄ = 7

So, first four terms of the sequence is 1, 3, 5, 7.

Problem 4 :

a_n = n/(n +1)

Solution :

a_n = n/(n +1)

a₄ = 4/(4 +1)

a₄ = 4/5

So, first four terms of the sequence is 1/2, 2/3, 3/4, 4/5.

Problem 5 :

a_n = 12 - 3n

Solution :

a_n = 12 - 3n

a₄ = 12 – 3(4)

= 12 – 12

a₄ = 0

So, first four terms of the sequence is 9, 6, 3, 0.

Problem 6 :

a_n = 4n/3

Solution :

a_n = 4n/3

a₄ = 4(4)/3

a₄ = 16/3

So, first four terms of the sequence is 4/3, 8/3, 4, 16/3.