What this chapter is about
A sequence is an ordered list of numbers following a definite rule, where each number is called a term. When we add the terms of a sequence together, we get a series. This chapter introduces you to the systematic study of patterns in numbers—how they grow, shrink, or change according to fixed rules.
You will focus mainly on two special types: arithmetic progressions (AP), where each term differs from the previous one by a constant, and geometric progressions (GP), where each term is a constant multiple of the previous one. These patterns appear everywhere—in simple interest calculations, population growth, the design of staircases, and even in music. Understanding sequences and series builds the foundation for calculus, where infinite series become essential tools.
By the end of this chapter, you should be able to identify whether a given sequence is an AP or GP, find any term of such sequences, calculate the sum of a specified number of terms, and solve practical problems involving these progressions.
Key ideas
- A sequence is a function from the set of natural numbers to the set of real numbers; we denote the terms as a₁, a₂, a₃, ... or simply {aₙ}.
- In an arithmetic progression (AP), the difference between consecutive terms is constant; this constant is called the common difference, denoted d. So aₙ₊₁ − aₙ = d for all n.
- In a geometric progression (GP), the ratio of consecutive terms is constant; this constant is called the common ratio, denoted r. So aₙ₊₁ / aₙ = r for all n, where aₙ ≠ 0.
- The arithmetic mean (AM) of two numbers a and b is (a + b)/2; it lies exactly midway between them on the number line.
- The geometric mean (GM) of two positive numbers a and b is √(ab); for positive numbers, GM ≤ AM always.
- A series is the sum of terms of a sequence; when the sequence is finite, the series has a definite sum, and when infinite, convergence must be checked.
- The sum of an infinite GP with |r| < 1 converges to a/(1 − r), where a is the first term.
Formulas and facts to remember
For an Arithmetic Progression with first term a and common difference d:
- nth term: aₙ = a + (n − 1)d — each term is the first term plus (n − 1) times the common difference.
- 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.
- Arithmetic mean of a and b: AM = (a + b)/2.
For a Geometric Progression with first term a and common ratio r:
- nth term: aₙ = a × r^(n − 1) — each term is the first term multiplied by r raised to (n − 1).
- Sum of first n terms: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1; Sₙ = na when r = 1.
- Sum of infinite GP (|r| < 1): S∞ = a/(1 − r).
- Geometric mean of two positive numbers a and b: GM = √(ab).
Relation between AM and GM:
- For any two positive numbers a and b: AM ≥ GM, with equality only when a = b.
Worked examples
Example 1: Finding terms of an AP
A farmer plants trees in a row. The first tree is 3 metres from the boundary, and each subsequent tree is 2 metres further from the previous one. Find the position of the 15th tree from the boundary.
Solution: Here a = 3 (first term), d = 2 (common difference), n = 15. Using aₙ = a + (n − 1)d: a₁₅ = 3 + (15 − 1) × 2 a₁₅ = 3 + 14 × 2 a₁₅ = 3 + 28 = 31 metres.
The 15th tree is 31 metres from the boundary.
Example 2: Sum of a GP
A ball is dropped from a height of 16 metres. After each bounce, it rises to half the previous height. Find the total distance travelled by the ball before it comes to rest.
Solution: The ball falls 16 m, then rises 8 m and falls 8 m, then rises 4 m and falls 4 m, and so on. Total distance = 16 + 2(8 + 4 + 2 + 1 + ...).
The series 8 + 4 + 2 + 1 + ... is an infinite GP with a = 8 and r = 1/2. Since |r| < 1, the sum converges: S∞ = a/(1 − r) = 8/(1 − 1/2) = 8/(1/2) = 16 metres.
Total distance = 16 + 2 × 16 = 16 + 32 = 48 metres.
Example 3: Inserting arithmetic means
Insert three arithmetic means between 4 and 20.
Solution: We need an AP with a₁ = 4, a₅ = 20, and three terms in between. So n = 5. Using aₙ = a + (n − 1)d: 20 = 4 + (5 − 1)d 20 = 4 + 4d 4d = 16 d = 4.
The five terms are: 4, 8, 12, 16, 20. The three arithmetic means are 8, 12, and 16.
Common mistakes
- Using the GP sum formula with r = 1 → when r = 1, all terms are equal, so Sₙ = na; the standard formula has division by zero.
- Confusing the nth term with the sum of n terms → aₙ gives one specific term, while Sₙ gives the total of all terms up to that point.
- Applying the infinite GP formula when |r| ≥ 1 → the series diverges; the formula a/(1 − r) only works when |r| < 1.
- Forgetting that r can be negative in a GP → a sequence like 3, −6, 12, −24, ... is a GP with r = −2; the signs alternate.
- Calculating GM of negative numbers directly using √(ab) → the standard GM formula applies to positive numbers; for negatives, interpret carefully or state it is undefined for real numbers.
Quick revision
- AP: constant difference between terms; nth term = a + (n − 1)d; sum = n/2 × (first + last).
- GP: constant ratio between terms; nth term = a × r^(n − 1); sum of infinite GP (|r| < 1) = a/(1 − r).
- AM of a and b is (a + b)/2; GM of positive a and b is √(ab); AM ≥ GM always.
- Check whether a sequence is AP by verifying aₙ₊₁ − aₙ is constant; check GP by verifying aₙ₊₁/aₙ is constant.
- To find the number of terms, use the nth term formula and solve for n.