Progressions (A.P., G.P., H.P.)

A progression is a sequence of numbers whose terms are arranged in a specific pattern.

Arithmetic Progression (A.P.): A sequence where the difference between consecutive terms is constant. This is called the common difference (d). Example: 3, 7, 11, 15... (d=4)
Geometric Progression (G.P.): A sequence where the ratio of any term to its preceding term is constant. This is called the common ratio (r). Example: 2, 6, 18, 54... (r=3)
Harmonic Progression (H.P.): A sequence of numbers whose reciprocals form an Arithmetic Progression. Example: 1/2, 1/5, 1/8... (Reciprocals 2, 5, 8... are in A.P.)

Let 'a' be the first term and 'd' be the common difference / 'r' be the common ratio.

Arithmetic Progression (A.P.):
- n-th term: \( t_n = a + (n-1)d \)
- Sum of n terms: \( S_n = \frac{n}{2}[2a + (n-1)d] \)
Geometric Progression (G.P.):
- n-th term: \( t_n = ar^{n-1} \)
- Sum of n terms: \( S_n = a \frac{(r^n - 1)}{(r - 1)} \) for \( r \neq 1 \)
- Sum of infinite terms (when \( |r| < 1 \)): \( S_\infty = \frac{a}{1-r} \)
Means: Arithmetic Mean (AM) \( \frac{a+b}{2} \), Geometric Mean (GM) \( \sqrt{ab} \). An important relation is \( AM \ge GM \).

For problems involving 3 terms in A.P., assume the terms as \( (a-d), a, (a+d) \).
For problems involving 3 terms in G.P., assume the terms as \( (a/r), a, (ar) \).
To solve H.P. problems, convert the sequence into an A.P. by taking the reciprocal of each term, solve using A.P. formulas, and then convert the result back by taking its reciprocal.

First, correctly identify the type of progression. Look for a common difference or a common ratio.
Be very careful with the conditions for the sum of an infinite G.P. formula (\( |r| < 1 \)).
The AM-GM inequality is a powerful tool for questions asking for the minimum or maximum value of expressions.