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Class 9 Predicting What Comes Next: Exploring Sequences and Progressions: cheat sheet
CBSE Mathematics. In an arithmetic progression each term is the last one plus a fixed number d. Find any term, or the sum of many terms, with two formulas.
Key ideas
- An AP has a first term a and a common difference d = second term − first term.
- d can be negative (a falling sequence) or a fraction.
- To check for an AP, the difference between every pair of neighbouring terms must be the same.
- The number of terms n must be a positive whole number; reject other solutions.
- Three numbers in an AP are easiest to write as a − d, a, a + d.
Remember
| nth term | aₙ = a + (n − 1)d |
| Sum of n terms | Sₙ = n/2 × (2a + (n − 1)d) |
| Sum from first and last term l | Sₙ = n/2 × (a + l) |
| A term from the sum | aₙ = Sₙ − Sₙ₋₁ |
Watch out
- Using nd instead of (n − 1)d.
- Taking d as first − second instead of second − first.
- Keeping a negative or fractional value of n.
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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