Arithmetic Progression (A.P) Basics

Study Notes

The general term (n^th term) of an A.P. is given by t_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference.
The sum of the first 'n' terms of an A.P is given by S_n = (2a + (n-1)d)/2.
In an A.P., the difference between any two consecutive terms is constant, which is called the common difference.
To determine if a sequence is an A.P., check if the difference between consecutive terms is constant.
An A.P can be represented as a sequence with a constant difference between terms. Examples include: 2k, 5k, 8k, 11k or 2/k, 5/k, 8/k, 11/k or 2+k, 5+k, 8+k, 11+k or 2-k, 5-k, 8-k, 11-k.
For a 3 term A.P., they can be expressed as: a, a+d, a+2d or a-d, a, a+d.
For a 4 term A.P., they can be expressed as: a-3d, a-d, a+d, a+3d.
For a 5 term A.P., they can be expressed as: a-2d, a-d, a, a+d, a+2d.
In an A.P., the difference between the last term and the second last term is equal to the common difference.
The difference between the n^th term and (n-1)^th term is equal to the common difference 'd', indicating the constant difference between terms.

We can insert 'n' arithmetic means between any two numbers 'a' and 'b' to form an A.P.
The common difference (d) for this series is calculated as d = (b+a)/(n-1)
The 'n' arithmetic means are: a + d, a + 2d, ..., a + nd
The sum of all arithmetic means between 'a' and 'b' can be calculated as Sam = n(a+b)/2, where 'n' is the number of arithmetic means.

The general term (n^th term) of a G.P. is given by t_n = ar^n-1, where 'a' is the first term and 'r' is the common ratio.
The sum of the first 'n' terms of a G.P is given by S_n = a(1-rⁿ)/(1-r).
If 'r' = 1, the sum of a G.P is given by S_n = na.
The sum of an infinite G.P. for -1 < r < 1 is S₀ = a/(1-r), where 'a' is the first term and 'r' is the common ratio.
In a G.P, the ratio between any two consecutive terms is constant, which is called the common ratio. To determine if a sequence is a G.P, check if the ratio between consecutive terms is constant .
A G.P can be represented as a sequence with a constant ratio between consecutive terms.
For a 3 term G.P, they can be expressed as: a, ar, ar² or a, a, ar
For a 4 term G.P, they can be expressed as: a/r, a, ar, ar³
For a 5 term G.P, they can be expressed as: a/r², a/r, a, ar, ar²
In G.P, the ratio between the n^th term and (n-1)^th term is equal to the common ratio.
The difference between S_n and S_n-1 is equal to the n^th term (T_n) of the G.P. Notably, T_n is equal to r (common ratio) multiplied by T_n-1 (previous term).
In any G.P, if a, b, and c are consecutive terms, then b² = ac. This can be further written as a/b = b/c.

The geometric mean for two numbers, 'a' and 'b', is given by (ab)^1/2.
The geometric mean for three numbers, 'a', 'b', and 'c', is given by (abc)^1/3.
The geometric mean for 'n' numbers (a, b, c, ..., a_n) is given by (a, b, c,..., a_n)^1/n.

We can insert 'n' geometric means between any two positive numbers 'a' and 'b' to form a G.P.
The common ratio (r) for this series is calculated as r =( b/a)^1/n.

The arithmetic mean (AM) of a set of numbers is always greater than or equal to the geometric mean (GM) of the same set of numbers. This inequality can be expressed as A.M ≥ G.M.

The following are some important formulas involving sigma notation:
- Σ_r=1ⁿ r = n(n+1)/2
- Σ_r=1ⁿ r² = n(n+1)(2n+1)/6
- Σ_r=1ⁿ r³ = (n(n+1)/2) ²