Library / Foundations: Maths / FMA-17
Foundations: Maths
Arithmetic series: sum of n terms
Total of the first n terms when each rises by a constant step.
- Ref
- FMA-17
- Rev
- v1
- Field
- Foundations: Maths
- Result
- Sn
a is the first term, d the common difference and n the number of terms.
Pair the first and last terms, the second and second-to-last, and so on: each pair has the same total, so the sum is the number of pairs times that total.
Calculator
- Last term a + (n − 1)d39
- Mean of the terms21
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The sum equals the number of terms times the average of the first and last terms.
- The sum of the first n whole numbers is n(n+1)/2, the case a = d = 1.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Number of terms | 3 to 100 | 21 | 20,100 | 20,080 | +10% | +20.5 % |
| d Common difference | 1 to 10 | 75 | 480 | 405 | +10% | +8.57 % |
| a First term | 1 to 20 | 190 | 380 | 190 | +10% | +1.43 % |
Sweep of Number of terms
| Number of terms | Sum of n terms |
|---|---|
| 3 | 21 |
| 15.13 | 472.7 |
| 27.25 | 1,512 |
| 39.38 | 3,140 |
| 51.5 | 5,356 |
| 63.63 | 8,160 |
| 75.75 | 11,550 |
| 87.88 | 15,530 |
| 100 | 20,100 |
Sweep one variable
Variables and how each one contributes
Sum of n terms
What the equation gives you. Everything else feeds it.
First term
The first term. Adds n times its value to the sum.
Common difference
Step between terms. Its effect grows with n²: more terms make the step far more important.
Number of terms
How many terms are added. The sum grows roughly with n².
Worked example with the values above
- Last term: 3 + 9 × 4 = 39.
- Average of first and last: 21.
- Sn = 10 × 21 = 210.
Assumptions and limits
- A constant common difference.
- n is a positive whole number.
Common mistakes
- Using n instead of n − 1 in the bracket.
- Mixing up the sum with the nth term.
Statistics and uncertainty
The sum grows roughly with n², so an error in the count n has double the relative effect on the total.
Where it is used
- Total stack or layer counts.
- Cumulative totals of a steadily changing quantity.
Maths you need
- Arithmetic sequences: nth term a + (n−1)d
- Algebra: brackets