Sequences: Determine the general term of the following sequences: A. 3, 7, 11, 15, 19, ... B. -3, 9, -27, 81, -243, ... C. (3)/(4) , (5)/(7) , (7)/(10) , (9)/(13) , (11)/(16) , …

Question

Sequences:

Determine the general term of the following sequences:

A. 3, 7, 11, 15, 19, ...

B. -3, 9, -27, 81, -243, ...

C.
`(3)/(4) , (5)/(7) , (7)/(10) , (9)/(13) , (11)/(16) , ...`

Answer 1

A.

• 3, 7, 11, 15, 19, ...

The first term is 3 and the common difference is 4.

The nth term is:

• aₙ = a₁ + (n - 1)d
• aₙ = 3 + 4(n - 1) = 4n - 1
• aₙ = 4n - 1

B.

• -3, 9, -27, 81, -243, ...

The first term is -3 and common ratio is -3.

The nth term is:

• aₙ = a₁rⁿ⁻¹
• aₙ = -3*(-3)ⁿ⁻¹ = (-3)°
• aₙ = (-3)ⁿ

C.

• 3/4, 5/7, 7/10, 9/13, 11/16, ...

The numerator and the denominator are both AP.

The numerator:

• b₁ = 3, d = 2

The nth term is:

• bₙ = 3 + 2(n - 1) = 2n + 1

The denominator:

• c₁ = 4, d = 3

The nth term is:

• cₙ = 4 + 3(n - 1) = 3n + 1

The nth term of the sequence is:

• aₙ = bₙ/cₙ
• aₙ = (2n + 1)/(3n + 1)