## Series of numbers

Sometimes the SAT will ask you to notice a pattern in a sequence of numbers.

-3, -1, 1, 3, 5

These numbers form what are known as a sequence. The one above is known as an **arithmetic sequence**, because each number increases by a fixed sum. So if we were to continue the sequence, we would start with 7, continuing to add by two (…7, 9, 11, 13…).

The little dots (…), by the way, mean that numbers came before (as in the case of ‘7’) or after (as in the case of ‘13’).

Now for a slightly trickier series:

2, 4, 8, 16, 32…256

Okay, that’s actually not that difficult either—each number is increasing by (x2). Notice, I’m not adding by a fixed number, the way I was doing above. When you multiply by a fixed number, you get a **geometric sequence**. Remember the dots just mean that you don’t write out all of the numbers, though the pattern continues. With this sequence, instead of writing 64, 128, and 256, I just replace them with dots and end with 512, which is the last number in the sequence.

Now that you’ve got the basics, let’s take it up a few notches.

Drum roll please: It’s the Challenge Problem!

## Challenge Problem

-1/128, 1/32, -1/8, 1/2…2048

In the sequence above, how many values are less than -1?

(A) Zero

(B) Two

(C) Three

(D) Five

(E) Six

The first thing to remember is that -1/128 and -1/8 are not less than -1. On a number line, they will both fall to the right of -1, because they are both closer to zero, and thus greater than -1. Of course a challenge problem wouldn’t be that straightforward, where all you have to do was count the numbers provided. In this case, there are a few missing numbers, which are key to getting this right.

To figure out these numbers, we have to unlock a pattern. Can you figure out what it is, and what the actual answer is (Hint: It’s not (B)!).

Leave a comment with your work and your answer, or let me know if you get stuck!

Hey, thanks for putting the time into this

Worked this out

the ratio is -4 knowing this you just use the formular a(-4)^n-1

where a=-2 because 1/2 . -4= -2

then find n using logs which gives you 6

you know that half of the numbers will be negative cause even power of -4 makes a positive term and odd powers of -4= even term

I follow you on most of that, but you didn’t quite get to the answer. So I want to say “good job”, but to be sure, what would you say your final answer is?

The answer is C (3 terms)

Good job, Abhishek! That’s the answer 🙂