Arithmetic Progression Class 10: Formulas & Examples
Use the nth-term and sum formulas for an arithmetic progression, with worked examples, a formula sheet and practice answers.
About 3 min readReviewed by Codex (AI content review)Syllabus NCERT 2026–27 reprint (selected topics)Updated 4/9/2026
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An arithmetic progression (AP) is a sequence with the same difference between every pair of consecutive terms. Before using an AP formula, subtract adjacent terms. A pattern that looks regular, such as 2, 4, 8, 16, is not an AP because its differences change.
AP formula sheet
| Quantity | Formula | Meaning |
|---|---|---|
| Common difference | d = a₂ − a₁ | Subtract an earlier term from the next one. |
| nth term | aₙ = a + (n − 1)d | a is the first term; n counts positions starting at 1. |
| Sum of first n terms | Sₙ = n[2a + (n − 1)d]/2 | Adds the first n terms, not the nth term alone. |
| Sum when last term l is known | Sₙ = n(a + l)/2 | Use when n and the first and last terms are known. |
Find a term and a position
The AP 7, 11, 15, …
- 1a = 7 and d = 4. The 20th term is 7 + 19 × 4 = 83.
- 2To locate 67, solve 67 = 7 + (n − 1)4. Then n − 1 = 15, so n = 16.
- 3To test 68, the same calculation gives n = 16.25. Positions are positive integers, so 68 is not a term.
Find a sum
Add the first 20 terms of 7, 11, 15, …
- 1The last of these terms is 83. Pair the first and last: 7 + 83 = 90.
- 2There are 10 such pairs, so S₂₀ = 10 × 90 = 900.
- 3The formula agrees: 20[14 + 19 × 4]/2 = 900.
Decreasing sequences
The AP 30, 27, 24, …
- 1d = 27 − 30 = −3. The tenth term is 30 + 9(−3) = 3.
- 2S₁₀ = 10(30 + 3)/2 = 165.
- 3Keep d negative throughout; do not turn a decreasing AP into an increasing one.
Practice with answers
- 1. Find the 15th term of 4, 9, 14, … Answer: 4 + 14 × 5 = 74.
- 2. Find the sum of the first 12 positive multiples of 3. Answer: 3 + 6 + … + 36 = 12(3 + 36)/2 = 234.
- 3. Is 52 a term of 5, 11, 17, …? Answer: no; solving gives n = 53/6, not a positive integer.
- 4. A student saves ₹10 in week 1 and ₹5 more each successive week. Find the first 8 weeks’ total. Answer: 8[20 + 7 × 5]/2 = ₹220.
Always label whether your answer is a single term or a total. The twentieth term in the first example is 83, but the sum of twenty terms is 900. Confusing aₙ with Sₙ is a common source of otherwise correct-looking answers.