Arithmetic Progression (A.P) Basics

Arithmetic Progression (A.P)

• 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.

Inserting Arithmetic Means

• 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.

Geometric Progression (G.P)

• 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.

Geometric Mean (GM)

• 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.

Inserting Geometric Means

• 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.

A.M. and G.M. Inequality

• 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.

Sigma Notation

• 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