SAT Sequence questions can be challenging. Can you address one of these problems in your blog?
Yes! Today's post focuses on a sequence problem. Sequence questions can be deceptively difficult – they are rarely as easy as 1, 2, 3 – so let's review a trickier one below.
2, -4, 8, …
The first term of the sequence above is 2, and every term after the first term is –2 times the preceding term. How many of the first 50 terms of this sequence are less than 100?
Okay, let's start filling in the sequence: -16, 32, -64, 128…
While you can certainly fill in the rest of the sequence, we're only at the 7th term and multiplying each term by -2 creates some annoying and time-consuming arithmetic. Getting to 50 terms might take us all day!
Do you have a shortcut for this SAT problem?
There are two keys to this problem: (1) Clearly understanding what the question entails, and (2) Recognizing the pattern of the sequence.
Let's start with (1). A common mistake with this problem is interpreting "less than 100" as only numbers between 0 and 99. But remember, "less than 100" means that all negative numbers are also relevant, not just small positive numbers.
Now that we've got (1), let's move on to (2). Since we multiply each term by -2, we can see that every other term (or half of the total terms) in the sequence will be negative. Out of 50 terms, 25 will be negative and therefore less than 100.
So is that the answer?
Almost! We still need to identify the positive terms that are less than 100. Of the first seven terms that we wrote above, we can see than three – 2, 8, and 32 – are positive and less than 100; as soon as we get to 128, we reach terms greater than 100. To calculate the answer, we just need to add the negative terms (25) to the positive terms less than 100 (3): 25 + 3 = 28. The correct answer is C.
Make sure to join me for the next installment of the Math Mechanic coming in October.