An arithmetic progression is a list where each term is the previous term plus the same number. Every AP question uses two formulas plus a few habits. Here they are, with the situation each one is for.
The definitions
- a = first term, d = common difference = any term minus the one before it, n = number of terms.
- A sequence is an AP only if d is the same everywhere: 2, 5, 8, 11 (d = 3) is; 1, 2, 4, 8 is not.
- General form: a, a + d, a + 2d, a + 3d, …
The nth term — for "find the 20th term" and "which term is 97?"
aₙ = a + (n − 1)d
Worked line: for 3, 7, 11, …, the 20th term is 3 + 19 × 4 = 79. Reverse worked line: which term of 3, 7, 11, … is 99? 3 + (n − 1)4 = 99 ⇒ n − 1 = 24 ⇒ n = 25. If n comes out non-integer, the number is not a term of the AP — and saying so is the answer.
The nth term from the end — for "the 5th term from the end"
Term from the end = l − (n − 1)d, where l is the last term.
Worked line: in 4, 9, 14, …, 254, the 10th term from the end is 254 − 9 × 5 = 209. (Or reverse the AP with d = −5 and use the ordinary formula.)
The sum of n terms — for "sum of the first 30 terms"
Sₙ = n/2 [2a + (n − 1)d]
Sₙ = n/2 (a + l) when the last term l is known.
Worked line: sum of the first 30 terms of 2, 5, 8, … is 30/2 [4 + 29 × 3] = 15 × 91 = 1365. Second form: sum of 5 + 10 + … + 100. Here n = 20 (from 100 = 5 + (n − 1)5), so S = 20/2 (5 + 100) = 1050.
The nth term from the sums — for "Sₙ is given, find aₙ"
aₙ = Sₙ − Sₙ₋₁
Worked line: if Sₙ = 3n² + 5n, then a₁ = S₁ = 8, and a₁₀ = S₁₀ − S₉ = 350 − 288 = 62. Also d = a₂ − a₁ = (S₂ − S₁) − S₁ = (22 − 8) − 8 = 6.
Choosing terms cleverly — for "three numbers in AP whose sum is …"
- Three terms: a − d, a, a + d (sum = 3a, so a is found immediately).
- Four terms: a − 3d, a − d, a + d, a + 3d (sum = 4a; common difference is 2d).
- Five terms: a − 2d, a − d, a, a + d, a + 2d.
Worked line: three numbers in AP have sum 24 and product 440. Then 3a = 24 ⇒ a = 8, and (8 − d)(8)(8 + d) = 440 ⇒ 64 − d² = 55 ⇒ d = ±3. Numbers: 5, 8, 11.
Useful one-liners
- Sum of the first n natural numbers: n(n + 1)/2.
- If a, b, c are in AP, then b = (a + c)/2 (b is the arithmetic mean).
- Adding or multiplying every term by the same number keeps it an AP.
- Number of terms between two multiples: multiples of 7 between 100 and 300 are 105, 112, …, 294 ⇒ n = (294 − 105)/7 + 1 = 28.
The three you will mix up
- (n − 1) in aₙ but n/2 in Sₙ. The nth term has n − 1 gaps; the sum averages the first and last term over n terms.
- Sₙ formula with l needs the actual last term, not the last term you happen to know.
- "Which term" questions must give an integer n. A non-integer means "not a term" — write that sentence and take the mark.