# Trevor Klee, Tutor

## Online and Boston-based GMAT, GRE, LSAT, and MCAT Prep.

Contact me

### What to know about arithmetic and geometric sequences

Note: this post is taken from my GRE Quant Bible. If you’d like fully explained practice questions to go along with this, get the book!

## How to recognize:

Whenever we’re adding up a long sequence of numbers

## What to know:

1. Occasionally, we’ll be asked to add up a lot of consecutive numbers, or numbers that are all separated by a common difference (e.g. adding up all the odd numbers). This is a really difficult task to do manually.

Fortunately, there’s a shortcut. The shortcut is to pair up the numbers: first with last, second with second to last, etc. Then we multiply the sum of each pair with the number of pairs.

Example:

If we want to add up all the multiples of 3 between 1 and 100, that’s the same thing as wanting to add up all the multiples of 3 between 3 and 99.

So, that’d be 3+6+9+…+93+96+99. Pairing these up, we’d get (3+99)+(6+96)+(9+93)…, which would each be equal to 102.

There are 33 multiples of 3 between 3 and 99 (as 3=31 and 99=333). So, there are 332=16.5 pairs, or 16 pairs. The 17th number, 51, does not have a pair (as the only number that pairs with 66 to equal 102 is 66 itself).

So the sum would be 16*102+51=1683.

2. We can also be asked to add or multiply geometric sequences. These are numbers that are not separated by a common difference, like being asked to add 1/2+1/4+1/6….

There’s no trick to calculating these. Instead, the best way to solve them is to do a few, find the pattern, and then extrapolate from the pattern.

## Takeaways:

1. When dealing with arithmetic sequences, pair them up, multiply the sum of the pairs by the number of pairs, and add back in any stragglers.