Sequences are ordered lists of numbers that follow a rule, and two types dominate school and college maths: arithmetic and geometric. They appear in Matric, FSc and A Level papers, and they model real situations like savings, loan repayments and population growth. This guide explains both types, their key formulas, and how to use them through fully worked examples.
What Is a Sequence?
A sequence is a list of numbers called terms, written \(a_1, a_2, a_3, \ldots\). The \(n\)th term is \(a_n\). A series is the sum of the terms of a sequence.
Arithmetic Sequences
In an arithmetic sequence (also called an arithmetic progression, A.P.), you add the same number each time. This number is the common difference, \(d\).
Example: \(3, 7, 11, 15, \ldots\) has \(a_1 = 3\) and \(d = 4\).
To find \(d\), subtract any term from the next one: \(d = a_2 - a_1\).
The \(n\)th Term
$$ a_n = a_1 + (n - 1)d $$
Sum of the First \(n\) Terms
$$ S_n = \frac{n}{2}\left[2a_1 + (n - 1)d\right] \qquad \text{or} \qquad S_n = \frac{n}{2}(a_1 + a_n) $$
Worked Example 1
The sequence \(5, 8, 11, 14, \ldots\) is arithmetic. Find (a) the 20th term, (b) the sum of the first 20 terms, (c) which term equals 101.
Step 1: Identify \(a_1 = 5\) and \(d = 8 - 5 = 3\).
(a) Use the \(n\)th term formula:
$$ a_{20} = 5 + (20 - 1)(3) = 5 + 57 = 62 $$
(b) Use the sum formula with \(a_1\) and \(a_n\):
$$ S_{20} = \frac{20}{2}(5 + 62) = 10 \times 67 = 670 $$
(c) Set \(a_n = 101\):
$$ 5 + (n - 1)(3) = 101 \quad \Rightarrow \quad 3(n - 1) = 96 \quad \Rightarrow \quad n - 1 = 32 \quad \Rightarrow \quad n = 33 $$
So 101 is the 33rd term.
Arithmetic Mean
The arithmetic mean of two numbers \(a\) and \(b\) is the middle term of an A.P.:
$$ A = \frac{a + b}{2} $$
Geometric Sequences
In a geometric sequence (geometric progression, G.P.), you multiply by the same number each time. This number is the common ratio, \(r\).
Example: \(2, 6, 18, 54, \ldots\) has \(a_1 = 2\) and \(r = 3\).
To find \(r\), divide any term by the previous one: \(r = \frac{a_2}{a_1}\).
The \(n\)th Term
$$ a_n = a_1 r^{n - 1} $$
Sum of the First \(n\) Terms
$$ S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad r \neq 1 $$
When \(r > 1\), the equivalent form \(S_n = \frac{a_1(r^n - 1)}{r - 1}\) keeps the numbers positive.
Worked Example 2
For the geometric sequence \(3, 6, 12, 24, \ldots\), find (a) the 8th term, (b) the sum of the first 8 terms.
Step 1: Identify \(a_1 = 3\) and \(r = \frac{6}{3} = 2\).
(a) Use the \(n\)th term formula:
$$ a_8 = 3 \times 2^{7} = 3 \times 128 = 384 $$
(b) Use the sum formula (with \(r > 1\)):
$$ S_8 = \frac{3(2^8 - 1)}{2 - 1} = 3(256 - 1) = 3 \times 255 = 765 $$
Check: \(3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 = 765\). Correct.
Geometric Mean
The geometric mean of two positive numbers \(a\) and \(b\) is
$$ G = \sqrt{ab} $$
For example, the geometric mean of 4 and 9 is \(\sqrt{36} = 6\), and \(4, 6, 9\) form a G.P. with \(r = 1.5\).
Infinite Geometric Series
If \(|r| < 1\), the terms get smaller and smaller, and the sum approaches a fixed value:
$$ S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1 $$
If \(|r| \geq 1\), the series has no finite sum.
Worked Example 3
Find the sum to infinity of \(12 + 6 + 3 + 1.5 + \cdots\).
Step 1: \(a_1 = 12\), \(r = \frac{6}{12} = \frac{1}{2}\). Since \(|r| < 1\), the sum exists.
Step 2: Apply the formula:
$$ S_\infty = \frac{12}{1 - \frac{1}{2}} = \frac{12}{\frac{1}{2}} = 24 $$
Worked Example 4: Recurring Decimals
Write \(0.\overline{7} = 0.777\ldots\) as a fraction.
Step 1: Express it as a series:
$$ 0.777\ldots = 0.7 + 0.07 + 0.007 + \cdots $$
Step 2: This is geometric with \(a_1 = 0.7\) and \(r = 0.1\):
$$ S_\infty = \frac{0.7}{1 - 0.1} = \frac{0.7}{0.9} = \frac{7}{9} $$
How to Tell Them Apart
- Differences constant? Arithmetic.
- Ratios constant? Geometric.
- Neither? It may be another type, such as quadratic (\(1, 4, 9, 16, \ldots\)).
Common Mistakes
- Using \(n\) instead of \(n - 1\) in the \(n\)th term formulas. The first term has no added \(d\) and no extra factor of \(r\).
- Getting the ratio upside down. Always divide a term by the one before it.
- Forgetting negative ratios. In \(4, -2, 1, -\frac{1}{2}, \ldots\), \(r = -\frac{1}{2}\).
- Using \(S_\infty\) when \(|r| \geq 1\). The formula only works for \(|r| < 1\).
Key Takeaways
- Arithmetic sequences add a common difference: \(a_n = a_1 + (n - 1)d\).
- Geometric sequences multiply by a common ratio: \(a_n = a_1 r^{n - 1}\).
- Use \(S_n = \frac{n}{2}(a_1 + a_n)\) for arithmetic sums and \(S_n = \frac{a_1(1 - r^n)}{1 - r}\) for geometric sums.
- An infinite geometric series converges to \(\frac{a_1}{1 - r}\) only when \(|r| < 1\).
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