8.16 Quick Revision - Arithmetic Progression
Core Definitions
AP: a, a+d, a+2d, a+3d, ...
a = first term
d = common difference = a(n) - a(n-1)
Essential Formulas
| # | Formula | Expression |
|---|
| 1 | nth term | a(n) = a + (n-1)d |
| 2 | Sum of n terms | S(n) = n/2 * [2a + (n-1)d] |
| 3 | Sum (using last term) | S(n) = n/2 * (a + l) |
| 4 | Number of terms | n = (l - a)/d + 1 |
| 5 | Common difference | d = (a(q) - a(p)) / (q - p) |
| 6 | nth term from end | l - (n-1)d |
| 7 | nth term from sum | a(n) = S(n) - S(n-1) |
| 8 | Arithmetic Mean | AM = (a + b) / 2 |
| 9 | n AMs between a, b | d = (b-a)/(n+1) |
| 10 | Sum of n AMs | n(a+b)/2 |
Standard Sums
| Series | Sum |
|---|
| 1 + 2 + 3 + ... + n | n(n+1)/2 |
| 2 + 4 + 6 + ... + 2n | n(n+1) |
| 1 + 3 + 5 + ... + (2n-1) | n^2 |
| 1^2 + 2^2 + ... + n^2 | n(n+1)(2n+1)/6 |
| 1^3 + 2^3 + ... + n^3 | [n(n+1)/2]^2 |
Key Properties
1. Three in AP: 2b = a + c
2. Equidistant: a(1)+a(n) = a(2)+a(n-1) = constant
3. Odd terms: Middle term = Average = S(n)/n
4. If S(n) = An^2 + Bn: d = 2A, first term = A + B
5. Adding/subtracting constant k: still AP, same d
6. Multiplying by k: still AP, new d = kd
Symmetric Selection
| Terms | Choose | CD |
|---|
| 3 | a-d, a, a+d | d |
| 4 | a-3d, a-d, a+d, a+3d | 2d |
| 5 | a-2d, a-d, a, a+d, a+2d | d |
Quick Reference Values
Sum 1 to 10 = 55 Sum 1 to 50 = 1275
Sum 1 to 20 = 210 Sum 1 to 100 = 5050
Common Mistakes to Avoid
1. n = (l-a)/d + 1 (don't forget the +1)
2. d can be negative (AP can decrease)
3. S(n) != a(n) (sum vs. term)
4. Non-integer n means the value is NOT a term of the AP
Back to Section 8.16