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Class 10 Arithmetic Progressions: cheat sheet
CBSE Mathematics. Lists that go up or down by the same amount each time: find any term, and the sum of the first n terms.
Key ideas
- In an AP, the common difference d = (any term) − (the term before it) is the same all the way along.
- d can be positive, negative or zero.
- The nth term counts n − 1 steps of d from the first term a.
- The sum of the first n terms is the number of terms times the average of the first and last terms.
- Three numbers a, b, c are in AP exactly when b − a = c − b, that is 2b = a + c.
- The nth term can be found from the sums: aₙ = Sₙ − Sₙ₋₁.
Remember
| nth term | aₙ = a + (n − 1)d |
| Sum of n terms | Sₙ = n ÷ 2 × [2a + (n − 1)d] |
| Sum with the last term l | Sₙ = n ÷ 2 × (a + l) |
| nth term from the end | l − (n − 1)d |
Watch out
- Using n instead of n − 1 in the nth-term formula.
- Mixing up aₙ (one term) and Sₙ (a total).
- Losing the minus sign when d is negative.
Worked example
How many terms of the AP 14, 18, 22, … must be taken to give a sum of 1,440?
Working: n/2 × (28 + (n − 1) × 4) = 1440; the positive solution is n = 24
Answer: 24 terms
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