Dear math experts,

I'm a parent with an 8th-grade math question.

We are working on patterns and sequences and writing the rule with it.

The patterns that went in 2s, 4s and 5s we got, but we are stuck on the pattern that doubles. We can't come up with the rule or equation that solves it.

With the first number being the term number and the second being the term, this is the pattern:

1-1
2-2
3-4
4-8
5-16
6-32
7-64
100-128

What is the rule for this?

Thanks

Tim
Parent

Here are two replies.

Hi Tim,

I think the last row should be 8 - 128. If this is the case then in the second column each number after the first is twice the number above it. That is

2 = 2 1
4 = 2 2
8 = 2 4
16 = 2 8
etc.

Thus the pattern, using exponential notation is

 1 1 1 2 2 1 21 3 2 2 = 4 22 4 2 4 = 8 23 5 2 8 = 16 24 6 2 16 = 32 25 7 2 32 = 64 26 8 2 64 = 128 27

Penny

Hi Tim,

You already understand the rule: it doubles.

So I assume your question is how to express it in a mathematical expression.

This is a geometric sequence, so instead of using multiplication, you use exponents. You said you knew how to do one that goes up in three's. I'd write that as
11, 14, 17, 20, ...

an = a0 + 3(n-1)

And read that aloud as "the nth term of the sequence is the first term
plus three times n minus one."

So I'd write your doubling sequence this way:
an = a0 (2n-1)

And I'd read that as "the nth term of the sequence is the first term times two to the power of n minus one."

Arithmetic sequence have a constant term (like the 3 above) multiplied by something indicating the number of terms. Geometric sequences (like the doubling one) have a constant term (like the 2 above) raised to an exponent indicating the number of terms.

Hope this helps,
Stephen La Rocque.