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GMAT Quant
Arithmetic Sequences
Find terms, count terms, and sum evenly spaced lists using the common difference.
Definition
An arithmetic sequence changes by a constant difference d.
Once you know the first term and d, every later term is determined.
Nth term
The nth term is a_n = a1 + (n - 1)d.
The n - 1 appears because moving from term 1 to term n takes n - 1 equal jumps.
Example: first term 7, difference 5, find term 9.
- Jumps
- 8
- Added amount
- 8*5 = 40
- Term 9
- 7 + 40 = 47
Number of terms
Count terms with N = (last - first)/d + 1.
The +1 counts the starting term. This works only when the stated last value actually belongs to the sequence.
Example: 4, 7, 10, ..., 31
- First
- 4
- Last
- 31
- Difference
- 3
- Terms
- (31-4)/3 + 1 = 10
Sum
Sum an evenly spaced list by pairing endpoints.
Sum = N(first + last)/2. The average of the list is (first + last)/2.
Continue the example.
- N
- 10
- First + last
- 35
- Sum
- 10*35/2 = 175
Mini application
Try this: 5, 9, 13, ..., 45. How many terms?
Check divisibility before scrolling.
Solution
There are 11 terms.
(45 - 5)/4 + 1 = 10 + 1 = 11.
Remember this
Arithmetic sequences are evenly spaced lists.
Use the difference to locate terms and the endpoint average to sum them.