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Solution

Many students find sequence notation intimidating, but it doesn’t have to be. Let’s rephrase each sequence in normal language.

Sn = Sn – 1 + is just saying that every term in S is equal to the term before it, plus 3/2.

An = An – 1 – 1.5 is just saying that every term in A is equal to the term before it, minus 1.5.

Of course, 3/2 and 1.5 are the same, which is a good clue that there’s probably some simple way to solve this problem without actually summing up a sequence.

Since S1 = 2 and every term in S is just 3/2 greater than the term before it, Sequence S begins like this: 2, 3.5, 5, 6.5, 8, 9.5, 11....

Since A1 =18.5 and every term in A is just 1.5 less than the term before it, Sequence A begins like this: 18.5, 17, 15.5, 14, 12.5, 11, 9.5....

At this point, it looks as though the two sequences in Column A and Column B are going to have a lot of terms in common! Remember, any common elements appearing in both A and B can just be canceled out.

You could just write out all 13 terms for each column, or you could “skip up” to S13 by noting that S13 is just going to be S1 plus 1.5, twelve times (since it takes twelve “jumps” to get from 1 to 13). 2 plus 18 is 20, so S13 = 20.

You can also “skip up” to the final term in A. To get to A13, take A1 and subtract 1.5 twelve times (since it takes twelve “jumps” to get from 1 to 13). 18.5 minus 18 is 0.5.
So,

Column A looks like this:	And Column B looks like this:
2, 3.5, 5, ... 18.5, 20	18.5, 17, 15.5.... 2, 0.5

That is, all the terms from 2 to 18.5 are held in common by both sets, so we can safely subtract them out. Here’s what’s left:

Column A	Column B
20	0.5

The correct answer is A.

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