Which strategy best explains how to solve this problem?

colin is preparing for a marathon by running in his neighborhood. the first week, he runs one block. the next week, he runs twice as many blocks as the first week (2 blocks). each week, he plans to run twice as many blocks as he ran the week before.

how many blocks will colin run by the end of the sixth week?

This is a geometric sequence because each term is twice the value of the previous term.  So this is what would be called the common ratio, which in this case is 2.  Any geometric sequence can be expressed as:

a(n)=ar^(n-1), a(n)=nth value, a=initial value, r=common ratio, n=term number

In this case we have r=2 and a=1 so

a(n)=2^(n-1)  so on the sixth week he will run:

a(6)=2^5=32

He will run 32 blocks by the end of the sixth week.

Now if you wanted to know the total amount he runs in the six weeks, you need the sum of the terms and the sum of a geometric sequence is:

s(n)=a(1-r^n)/(1-r)  where the variables have the same values so

s(n)=(1-2^n)/(1-2)

s(n)=2^n-1 so 

s(6)=2^6-1

s(6)=64-1

s(6)=63 blocks

So he would run a total of 63 blocks in the six weeks.