8.17 Quick Revision - Geometric Progression
Core Definitions
GP: a, ar, ar^2, ar^3, ...
a = first term (a != 0)
r = common ratio (r != 0) = a(n) / a(n-1)
Essential Formulas
| # | Formula | Expression |
|---|
| 1 | nth term | a(n) = a * r^(n-1) |
| 2 | Sum of n terms (r != 1) | S(n) = a(r^n - 1)/(r - 1) or a(1 - r^n)/(1 - r) |
| 3 | Sum of n terms (r = 1) | S(n) = na |
| 4 | Sum to infinity (|r| < 1) | S = a / (1 - r) |
| 5 | Geometric Mean | GM = sqrt(a * b) |
| 6 | n GMs between a, b | r = (b/a)^(1/(n+1)) |
| 7 | Product of n GMs | (ab)^(n/2) |
| 8 | nth term from end | l / r^(n-1) |
| 9 | nth term from sum | a(n) = S(n) - S(n-1) |
Key Properties
1. Three in GP: b^2 = ac
2. Equidistant product: a(1)*a(n) = a(2)*a(n-1) = constant
3. Logs of GP form an AP: log(a), log(ar), log(ar^2) -> AP with d=log(r)
4. Product of n terms (odd n): (middle term)^n
5. Multiplying by constant k: still GP, same ratio r
6. Reciprocals of GP: also a GP with ratio 1/r
7. Powers: a^k, (ar)^k, (ar^2)^k -> GP with ratio r^k
AM-GM Relationship
For positive numbers a, b:
AM >= GM >= HM
(a+b)/2 >= sqrt(ab) >= 2ab/(a+b)
AM * HM = GM^2
Equality when a = b
Symmetric Selection
| Terms | Choose | CR |
|---|
| 3 | a/r, a, ar | r |
| 4 | a/r^3, a/r, ar, ar^3 | r^2 |
| 5 | a/r^2, a/r, a, ar, ar^2 | r |
Recurring Decimals
0.aaa... = a/9
0.ababab... = ab/99
0.abcabc... = abc/999
Bouncing Ball Formula
Total distance = h(1+r)/(1-r)
where h = drop height, r = bounce ratio
AP vs GP Quick Comparison
AP: Constant difference, a(n) = a + (n-1)d, linear growth
GP: Constant ratio, a(n) = a*r^(n-1), exponential growth
AP sum: n/2[2a+(n-1)d]
GP sum: a(r^n-1)/(r-1)
AP mean: (a+b)/2
GP mean: sqrt(ab)
AP condition: 2b = a+c
GP condition: b^2 = ac
Common Mistakes to Avoid
1. Sum to infinity exists ONLY when |r| < 1
2. GP terms cannot be zero
3. r can be negative (alternating signs)
4. Don't confuse AP and GP formulas
5. For negative r: even powers positive, odd powers negative
Back to Section 8.17