Progression: A.P. and G.P.

A sequence is a set of numbers arranged in a specific order. When the terms of a sequence are added together, it is called a series. In the JAMB syllabus, two main types of progressions are studied: Arithmetic Progression (A.P.) and Geometric Progression (G.P.).

1. Arithmetic Progression (A.P.)

An A.P. is a sequence where each term after the first is obtained by adding a constant value called the common difference (d). If a is the first term, the sequence is: a, a+d, a+2d, ...

• The nth term (T_n): T_n = a + (n - 1)d
• Common Difference (d): d = T_n - T_n-1
• Sum of the first n terms (S_n): S_n = n/2 [2a + (n - 1)d] OR S_n = n/2 [a + l], where l is the last term.

Example: Find the 10th term of 2, 5, 8...
Here a = 2, d = 5 - 2 = 3.
T₁₀ = 2 + (10 - 1)3 = 2 + 27 = 29.

2. Geometric Progression (G.P.)

A G.P. is a sequence where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio (r).

• The nth term (T_n): T_n = ar^n-1
• Common Ratio (r): r = T_n / T_n-1
• Sum of the first n terms (S_n):
  S_n = a(rⁿ - 1) / (r - 1) for r > 1
  S_n = a(1 - rⁿ) / (1 - r) for r < 1

3. Sum to Infinity (S_∞)

For a G.P. where the common ratio r is a fraction (i.e., |r| < 1), the sum of the terms as n approaches infinity is finite. This is called the sum to infinity.

• Formula: S_∞ = a / (1 - r)

Example: Find the sum to infinity of 1, 1/2, 1/4...
Here a = 1, r = 1/2.
S_∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2.