Sequences Exercises

Answer

The general term a_n is the decimal expansion of π to n decimal places.

This type of sequence, called a decimal expansion, is probably the most common type of sequence you'll see where the general term is described verbally instead of by a formula. It's easy to understand, but there isn't a simple formula for the general term.

Lucky for us, most sequences you see in calculus classes will be described by mathematical formulas because these sequences are easier to work with.

You probably encountered formulas a long time ago. Finding the n^th term of a sequence is the same as evaluating a mathematical formula. If you don't remember, it's just like riding a bicycle. It'll come back quick.

Answer

This sequence is called the harmonic sequence, and will show up a lot.

Answer

a_n = 2n + 1

a₁ = 2(1) + 1 = 3

a₂ = 2(2) + 1 = 5

a₃ = 2(3) + 1 = 7

a₄ = 2(4) + 1 = 9

This sequence gives the collection of all odd numbers starting at 3.

Answer

a_n = n³

a₁ = (1)³ = 1

a₂ = (2)³ = 8

a₃ = (3)³ = 27

a₄ = (4)³ = 64

Answer

a_n = 6 – n

a₁ = 6 – 1 = 5

a₂ = 6 – 2 = 4

a₃ = 6 – 3 = 3

a₄ = 6 – 4 = 2

This is the countdown sequence beginning from 6. (Okay, we made that name up.)

Answer

a_n = n² + n

a₁ = (1)² + 1 = 2

a₂ = (2)² + 2 = 6

a₃ = (3)² + 3 = 12

a₄ = (4)² + 4 = 20