The general term an 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 nth 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.
Example 2
Evaluate the first 4 terms of the sequence, starting with n = 1.
Answer
This sequence is called the harmonic sequence, and will show up a lot.
Example 3
Evaluate the first 4 terms of the sequence, starting with n = 1.
an = 2n + 1
Answer
an = 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.
Example 4
Evaluate the first 4 terms of the sequence, starting with n = 1.
an = n3
Answer
an = n3
a1 = (1)3 = 1
a2 = (2)3 = 8
a3 = (3)3 = 27
a4 = (4)3 = 64
Example 5
Evaluate the first 4 terms of the sequence, starting with n = 1.
an = 6 – n
Answer
an = 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.)
Example 6
Evaluate the first 4 terms of the sequence, starting with n = 1.
