Learning Objectives
- Identify arithmetic progressions and find the common difference
- Find the nth term of an AP
- Find the sum of first n terms of an AP
- Solve real-life problems using APs
Key Concepts
Arithmetic Progression (AP)
An AP is a sequence in which each term after the first is obtained by adding a fixed number called the common difference (d) to the preceding term.
General form: a, a + d, a + 2d, a + 3d, ...
where a = first term, d = common difference.
nth Term of an AP
aₙ = a + (n - 1)d
This is also called the general term or last term (l) when n is the total number of terms.
Sum of First n Terms
Sₙ = n/2 [2a + (n - 1)d]
Or equivalently: Sₙ = n/2 [a + l], where l is the last term.
Note: The nth term can also be found as aₙ = Sₙ - Sₙ₋₁ (for n ≥ 2).
Properties of AP
- If a constant is added to or subtracted from each term, the resulting sequence is also an AP with the same common difference.
- If each term is multiplied or divided by a non-zero constant k, the resulting sequence is an AP with common difference kd or d/k.
- Three numbers a, b, c are in AP if and only if 2b = a + c. Here b is the arithmetic mean of a and c.
Summary
An arithmetic progression is a sequence with a constant difference between consecutive terms. The nth term and sum formulas allow efficient computation without listing all terms. APs model many real-world situations such as savings plans, seating arrangements, and distance-time problems at constant speed.
Important Terms
- Common Difference (d)
- The constant value added to each term to get the next term; d = aₙ - aₙ₋₁
- General Term (aₙ)
- The nth term of the AP: a + (n-1)d
- Arithmetic Mean
- The middle value of three terms in AP; AM of a and b is (a+b)/2
Quick Revision
- d = a₂ - a₁ = a₃ - a₂ = ... (constant for all consecutive pairs)
- aₙ = a + (n - 1)d
- Sₙ = n/2 [2a + (n-1)d] = n/2 [a + l]
- aₙ = Sₙ - Sₙ₋₁ for n ≥ 2
- If d > 0, AP is increasing; if d < 0, AP is decreasing; if d = 0, all terms are equal