Comprehensive Calculator Guide
An arithmetic sequence adds the same amount, the common difference, at every step, as in 2, 5, 8, 11. The nth term is a₁ + (n − 1) × d and the sum of the first n terms is n × (a₁ + aₙ) ÷ 2. For 2, 5, 8, … the 10th term is 29 and the first ten terms add up to 155.
Term and sum
Reaching the 10th term from the first takes nine steps, not ten, hence (n − 1). The difference can be negative (4, 2, 0, −2) or zero. To find a term's position, rearrange: n = (aₙ − a₁) ÷ d + 1; if n is not a whole number, the value is not in the sequence.
The sum is the number of terms times the average of the first and last, the trick the young Gauss used to add 1 to 100: fifty pairs of 101, which is 5,050.
Where they appear
Any fixed increase repeated: savings that grow by a set amount each month, theatre rows that gain two seats per row, a fare with a fixed charge per kilometre. If the increase is a percentage instead, such as a 5% annual raise, use a geometric sequence.
How to Use
- 01Enter the first term.
- 02Enter the common difference, positive or negative.
- 03Enter n, the position you want (a whole number from 1).
- 04Read the nth term and the sum of the first n terms.
Formula Used
aₙ = a₁ + (n − 1) × d | Sₙ = n × (a₁ + aₙ) ÷ 2Practical Examples
Example 1: 2, 5, 8, …
10th term = 2 + 9 × 3 = 29; sum of ten terms = 155.
Example 2: rising savings
SAR 100 in month one, SAR 50 more each month: SAR 650 in month 12 and SAR 4,500 over the year.
Example 3: theatre seats
20 seats in row one, two more per row, 15 rows: 48 in the last row, 510 in total.
Important Tips
- Check the difference between several pairs of terms; if it varies, the sequence is not arithmetic.
- 1 + 2 + … + n = n × (n + 1) ÷ 2.
Common Mistakes to Avoid
- Multiplying d by n instead of n − 1.
- Using this for growth by a fixed percentage.
Frequently Asked Questions
- How do I find the nth term?
- aₙ = a₁ + (n − 1) × d: the 20th term of 5, 9, 13 is 5 + 19 × 4 = 81.
- How do I add up an arithmetic sequence?
- Multiply the number of terms by the sum of the first and last, then halve: 1 to 100 gives 5,050.
- How do I find a term's position?
- n = (value − a₁) ÷ d + 1.
- Arithmetic or geometric?
- Arithmetic adds a fixed amount; geometric multiplies by a fixed ratio.