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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.


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.


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.

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