An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is called the common difference (d).
General form:
a, a + d, a + 2d, a + 3d, ...
where
a = first termd = common differencenth term formula:
Tn = a + (n - 1)d
Sum of first n terms:
Sn = n/2 × [2a + (n - 1)d]
a = 3, d = 4.a = 5, d = 3, n = 20.A Geometric Progression is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).
General form:
a, ar, ar2, ar3, ...
where
a = first termr = common rationth term formula:
Tn = a × rn-1
The formula for the sum of the first n terms depends on the value of the common ratio r:
r ≠ 1)Sn = a × (rn - 1) / (r - 1)
This formula works for any r ≠ 1, whether r > 1 or r < 1.
r < 1)Sn = a × (1 - rn) / (1 - r)
This is algebraically equivalent and preferred when r < 1 because the numerator stays positive.
r = 1Sn = n × a
If the common ratio is 1, all terms are equal to a, so the sum is just n times a.
| Case | Formula | Notes |
|---|---|---|
r > 1 |
Sn = a × (rn - 1) / (r - 1) | Use as is |
0 < r < 1 |
Sn = a × (1 - rn) / (1 - r) | Preferred for positive numerator |
r = 1 |
Sn = n × a | All terms equal |
a = 2, r = 3.a = 1, r = 2, n = 5.a = 5, r = 1/2, and |r| < 1.
If |r| < 1 and the series is infinite, the sum is:
S∞ = a / (1 - r)
This formula does not apply if |r| ≥ 1.