Sequences Exercises

Answer

If we ignore the signs we see the sequence

1, 2, 4, 8, 16,...

These are the powers of 2 starting at n = 0. To account for the signs we need to multiply by (-1)^{n + 1}. Starting at n = 0, the general term is

a_n = (-1)^{n + 1}2ⁿ.

If we start instead at n = 1 the general term is

a_n = (-1)ⁿ2^{n – 1}.

Answer

If we start at n = 1 the numerator is n and the denominator is 5. The general term is . There's no reason to start at n = 0 because that would only make things more complicated.

Answer

If we look at the numerator first we see the sequence n! starting at n = 1. Then the denominator is 2n – 1. Starting at n = 1, the general term is

If instead we look at the denominator first we might see the sequence (2n + 1) starting at n = 0. Then the numerator is (n + 1)!. Starting at n = 0, the general term is

Answer

This is the sequence a_n = 3n starting at n = 0.

Answer

If we look at the denominator first we see the sequence (2n)!, starting at n = 1. Taking signs into account, the general term is

If we look at the signs first we might see the sequence (-1)ⁿ starting at n = 0. Then the general term is

Be Careful: When dealing with factorials, be careful with your parentheses. In particular,

(2n)! ≠ 2n!

For example, if n = 4 then

(2n)! = 8! = 40320

but

2n! = 2(4!) = 48.