Many natural and mathematical phenomena unfold step by step, and sequences provide the language to describe such ordered lists of numbers. A sequence is an arrangement of numbers in a definite order, and the sum of the terms of a sequence is called a series. Two types of sequences dominate the chapter: arithmetic progressions and geometric progressions.
The chapter begins with the definitions of sequences and series, and the notation for their terms. It then develops the arithmetic progression, with its nth term and the sum of the first n terms, and the geometric progression, with its general term, sum formulas, and the infinite geometric series. The chapter also introduces the arithmetic mean and geometric mean, and the relationship between them.
Arithmetic and geometric progressions appear throughout mathematics and science: loan payments and simple interest follow arithmetic patterns, while population growth, radioactive decay, and compound interest follow geometric patterns. The formulas developed here are also prerequisites for limits and the theory of series studied in later mathematics.
A sequence is a function whose domain is the set of natural numbers. A sequence is denoted by a1, a2, a3, ..., where an is the nth term. A sequence can be finite or infinite.
A series is the sum of the terms of a sequence. The sum of the first n terms of a sequence is denoted by Sn:
Sn = a1 + a2 + a3 + ... + an
If a sequence has a clear pattern, it can be specified by its general term an. For example, the sequence 1, 4, 9, 16, ... has an = n^2.
A sequence is an arithmetic progression if the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by d. If the first term is a, the sequence is:
a, a + d, a + 2d, a + 3d, ...
an = a + (n - 1)d
Sn = n/2 [2a + (n - 1)d] = n/2 (a1 + an)
A sequence is a geometric progression if the ratio of any two consecutive terms is constant. This constant ratio is called the common ratio, denoted by r. If the first term is a, the sequence is:
a, ar, ar^2, ar^3, ...
an = a r^(n-1)
For r not equal to 1: Sn = a(1 - r^n)/(1 - r) = a(r^n - 1)/(r - 1)
For r = 1, Sn = na.
If |r| < 1, the sum of the infinite geometric series is:
S(infinity) = a/(1 - r)
If |r| >= 1, the infinite geometric series does not converge to a finite sum.
The arithmetic mean of two positive numbers a and b is:
A = (a + b)/2
If a, A, b are in AP, then A is the arithmetic mean of a and b.
The geometric mean of two positive numbers a and b is:
G = sqrt(ab)
If a, G, b are in GP, then G is the geometric mean of a and b.
For positive numbers a and b:
A >= G
That is, (a + b)/2 >= sqrt(ab), with equality when a = b. This is a fundamental inequality.
Some important sums that are frequently used:
Sum of first n natural numbers: 1 + 2 + 3 + ... + n = n(n + 1)/2
Sum of squares of first n natural numbers: 1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1)/6
Sum of cubes of first n natural numbers: 1^3 + 2^3 + ... + n^3 = [n(n + 1)/2]^2
These formulas are proven by induction and are used to evaluate many series.
Finding the nth Term of an AP. Consider the arithmetic progression 7, 12, 17, 22, ... Here the first term is a = 7 and the common difference is d = 5, since each term exceeds the previous one by 5. The 20th term is obtained from the formula an = a + (n - 1)d, which gives a20 = 7 + 19 x 5 = 7 + 95 = 102. Notice that the 20th term uses 19 multiples of the common difference, not 20, because the first term itself is a1 and carries no added difference. This single detail is the most frequent source of error in the whole chapter, so always write down a and d before applying the formula.
Finding the Sum of an AP. For the same progression, the sum of the first 20 terms is S20 = n/2 [2a + (n - 1)d] = 20/2 [2(7) + 19(5)] = 10 [14 + 95] = 10 x 109 = 1090. An alternative is to use the second form Sn = n/2 (a1 + an). Here the last term is 102, so S20 = 20/2 (7 + 102) = 10 x 109 = 1090, which agrees perfectly with the first calculation. Choosing between the two forms depends on whether the last term is known directly; when it is not, the first form is safer because it needs only a, d and n.
Infinite Geometric Series. Consider the geometric progression with a = 3 and r = 1/3, giving the terms 3, 1, 1/3, 1/9, ... Since |r| = 1/3 < 1, the infinite series converges, and its sum is S = a/(1 - r) = 3/(1 - 1/3) = 3/(2/3) = 9/2 = 4.5. Geometrically, the partial sums 3, 4, 4.333, 4.444, ... approach 4.5 but never reach or overshoot it. This example illustrates why the condition |r| < 1 is so important: if the common ratio were greater than or equal to 1 in absolute value, the terms would not shrink and the series would grow without bound, making a finite sum impossible. The same logic underlies the divergence of a geometric series when |r| >= 1.
| Feature | Arithmetic Progression | Geometric Progression |
|---|---|---|
| Defined by | Common difference d | Common ratio r |
| Consecutive relation | a_(n+1) - an = d | a_(n+1)/an = r |
| nth term | a + (n-1)d | a r^(n-1) |
| Sum of n terms | n/2 [2a + (n-1)d] | a(1 - r^n)/(1 - r), r not 1 |
| Mean of a, b | (a + b)/2 | sqrt(ab) |
| Series | Formula |
|---|---|
| 1 + 2 + 3 + ... + n | n(n + 1)/2 |
| 1^2 + 2^2 + ... + n^2 | n(n + 1)(2n + 1)/6 |
| 1^3 + 2^3 + ... + n^3 | [n(n + 1)/2]^2 |
| Infinite GP ( | r |
Sequences and series organise the idea of growth by steps and by ratios. Arithmetic progressions capture constant addition, and geometric progressions capture constant multiplication, and together they model a vast range of natural and economic phenomena. The nth term and sum formulas, the arithmetic and geometric means, and the convergence condition of the infinite geometric series are the essential tools of this chapter. The relationship A >= G is a fundamental inequality with wide applications. These foundational ideas prepare students for limits, infinite series, and the deeper analysis encountered in higher mathematics and in applications across the sciences.