Progressions and Series formulas
At a glance
AP: tₙ = a + (n − 1)d Sₙ = n/2 × (a + l) = average × count GP: tₙ = a·rⁿ⁻¹ Sₙ = a(rⁿ − 1)/(r − 1) S∞ = a/(1 − r), |r| < 1 AM ≥ GM ≥ HM GM² = AM × HM middle term: AP 2b = a + c, GP b² = ac, HP b = 2ac/(a + c) Σk = n(n + 1)/2 Σk² = n(n + 1)(2n + 1)/6 Σk³ = (Σk)² 1/(k(k + d)) = (1/d)(1/k − 1/(k + d))
Arithmetic progressions: equal steps
nth term: tₙ = a + (n − 1)d step from any two terms: d = (tₘ − tₖ) / (m − k) number of terms from a to l: n = (l − a)/d + 1
Shortcut: Three terms in AP: a − d, a, a + d (their sum is 3a, so a pops out at once). Four terms: a − 3d, a − d, a + d, a + 3d, with step 2d.
Sum of an AP: flip and add
Sₙ = n/2 × (a + l) Sₙ = n/2 × [2a + (n − 1)d] average of an AP = (first + last)/2 = middle term
Shortcut: Terms equally far from the two ends always add to the same total: t₁ + tₙ = t₂ + tₙ₋₁ = … And with an odd number of terms, the sum is simply n × the middle term.
Geometric progressions: equal ratios
nth term: tₙ = a·rⁿ⁻¹ sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1 if r = 1: Sₙ = n·a
Shortcut: Three terms in GP: a/r, a, ar. Their product is a³, so a product of 216 tells you the middle term is 6. For any three consecutive terms, middle² = first × third.
Infinite GP: sums that stop growing
S∞ = a / (1 − r), only when −1 < r < 1
Shortcut: 0.363636… = 0.36 + 0.0036 + … = 0.36/(1 − 0.01) = 36/99 = 4/11.
AM, GM and HM
AM ≥ GM ≥ HM (equal only when a = b) GM² = AM × HM (for two numbers) 4 and 16: AM 10, GM 8, HM 6.4, and 10 × 6.4 = 64 = 8²
| Progression | a, b, c in it when | n means between a and b |
|---|---|---|
| AP | 2b = a + c | step d = (b − a)/(n + 1) |
| GP | b² = ac | ratio r = (b/a)^(1/(n + 1)) |
| HP (reciprocals in AP) | b = 2ac/(a + c) | insert APs between 1/a and 1/b, then flip |
An HP has no sum formula: turn it into an AP of reciprocals and work there.
Shortcut: Fixed sum: the product is largest when the parts are equal. Fixed product: the sum is smallest when the parts are equal. And x + 1/x ≥ 2 for every positive x.
Special sums: 1 + 2 + … + n, squares, cubes
1 + 2 + … + n = n(n + 1)/2 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 1³ + 2³ + … + n³ = [n(n + 1)/2]² 1 + 3 + 5 + … (n odd numbers) = n² 2 + 4 + … + 2n = n(n + 1)
Telescoping and arithmetic-geometric series
1/(k(k + d)) = (1/d) × (1/k − 1/(k + d)) 1/(1·2) + 1/(2·3) + … + 1/(n(n + 1)) = 1 − 1/(n + 1) = n/(n + 1)
Shortcut: 1/(√k + √(k + 1)) = √(k + 1) − √k. So the sum from k = 1 to 99 is √100 − √1 = 9.
Formulas stick when you use them
The Progressions and Series lesson has worked problems and quick checks for every formula here. Free to start.
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