What is a sequence in mathematics?

A sequence is an ordered list of numbers called terms, denoted a₁, a₂, a₃, …, aₙ, … . It is finite if it has a last term and infinite otherwise.

What is the difference between an explicit and a recursive formula?

An explicit formula expresses aₙ directly in terms of n (e.g., aₙ = 2n+1). A recursive formula gives aₙ in terms of one or more previous terms plus initial values (e.g., a₁ = 1, aₙ = 2aₙ₋₁ + 1).

What is summation (sigma) notation?

Summation notation uses the Greek letter Σ to compress a sum: ∑ᵢ₌₁ⁿ aᵢ = a₁ + a₂ + … + aₙ. The index i runs from the lower bound (1) to the upper bound (n).

What are the power-sum formulas for sigma notation?

∑ᵢ₌₁ⁿ i = n(n+1)/2, ∑ᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6, and ∑ᵢ₌₁ⁿ i³ = [n(n+1)/2]². For a constant c, ∑ᵢ₌₁ⁿ c = nc.

Sequences and Summation Notation

This tutorial covers mathematical sequences and summation (sigma) notation. It explains explicit and recursive formulas, the rules of sigma notation, common power-sum formulas, and includes six worked examples and three practice exercises with full solutions.

Introduction to Sequences

A sequence is an ordered list of numbers called terms. A sequence is finite if it has a last term and infinite if it continues indefinitely. Terms are usually written with subscripts:

\[ a_1,\ a_2,\ a_3,\ \ldots,\ a_n,\ \ldots \]

where \(a_n\) is the n-th term.

Example 1: The Even Numbers

Question: For the sequence of positive even numbers \( 2, 4, 6, 8, 10, \ldots \), find the general term \(a_n\) and the 20th term.

View solution

Each term is twice its index: \(a_n = 2n\). Hence \(a_1 = 2\), \(a_2 = 4\), \(a_3 = 6\), and so on. The 20th term is

\[ a_{20} = 2(20) = 40. \]

Example 2: A Reciprocal-Odd Sequence

Question: Write the general term of the sequence

\[ \frac{1}{3},\ \frac{1}{5},\ \frac{1}{7},\ \frac{1}{9},\ \frac{1}{11}, \ldots \]

View solution

The denominators are the odd integers starting at 3: \(3, 5, 7, 9, 11, \ldots\), which are \(2n+1\) for \(n = 1, 2, 3, \ldots\). Thus

\[ a_n = \frac{1}{2n+1}. \]

Check: \(a_1 = 1/3\), \(a_2 = 1/5\), \(a_3 = 1/7\) ✓

Example 3: An Explicit Formula with Alternating Sign

Question: Evaluate the first four terms of the sequence \( a_n = 2n + (-1)^n \).

View solution

\(a_1 = 2(1) + (-1)^1 = 2 - 1 = 1\)
\(a_2 = 2(2) + (-1)^2 = 4 + 1 = 5\)
\(a_3 = 2(3) + (-1)^3 = 6 - 1 = 5\)
\(a_4 = 2(4) + (-1)^4 = 8 + 1 = 9\)

The first four terms are \(1, 5, 5, 9\).

Example 4: A Recursively Defined Sequence

Question: Find the first five terms of the sequence defined by

\[ a_1 = 1,\quad a_2 = 3,\quad a_n = 2a_{n-1} - a_{n-2}\ \text{ for } n > 2. \]

View solution

A recursive definition uses earlier terms to compute later ones:

\(a_1 = 1\) (given)
\(a_2 = 3\) (given)
\(a_3 = 2a_2 - a_1 = 2(3) - 1 = 5\)
\(a_4 = 2a_3 - a_2 = 2(5) - 3 = 7\)
\(a_5 = 2a_4 - a_3 = 2(7) - 5 = 9\)

The first five terms are 1, 3, 5, 7, 9 — this is the arithmetic sequence \(a_n = 2n - 1\).

Summation (Sigma) Notation

The sum of consecutive terms \(a_1 + a_2 + \cdots + a_n\) is written compactly with the Greek letter sigma:

\[ \sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n \]

Here \(i\) is the index of summation, \(1\) is the lower bound, and \(n\) is the upper bound.

Useful Sigma Rules and Power-Sum Formulas

Constant: \( \displaystyle\sum_{i=1}^{n} c = nc \)
Linearity: \( \displaystyle\sum_{i=1}^{n}(ca_i + b) = c\sum_{i=1}^{n}a_i + nb \)
Sum of integers: \( \displaystyle\sum_{i=1}^{n} i = \frac{n(n+1)}{2} \)
Sum of squares: \( \displaystyle\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \)
Sum of cubes: \( \displaystyle\sum_{i=1}^{n} i^3 = \left[\frac{n(n+1)}{2}\right]^2 \)

Example 5: Sum of the First 7 Even Numbers

Question: Find the sum of the first seven positive even numbers: \(2, 4, 6, 8, 10, 12, 14, \ldots\)

View solution

Add the first 7 terms directly:

\[ 2 + 4 + 6 + 8 + 10 + 12 + 14 = 56 \]

Equivalently, using the sum of the first \(n\) integers:

\[ \sum_{i=1}^{7} 2i = 2\sum_{i=1}^{7} i = 2\cdot\frac{7\cdot 8}{2} = 56. \]

Example 6: Sum of a Rational Expression

Question: Evaluate \( \displaystyle\sum_{i=1}^{5} \frac{2i+1}{i} \).

View solution

Substitute \(i = 1, 2, 3, 4, 5\):

\[ S = \frac{3}{1} + \frac{5}{2} + \frac{7}{3} + \frac{9}{4} + \frac{11}{5} \]

Use the common denominator 60:

\[ S = \frac{180 + 150 + 140 + 135 + 132}{60} = \frac{737}{60}. \]

\( S = \dfrac{737}{60} \approx 12.2833 \).

Example 7: Using the Power-Sum Formulas

Question: Evaluate \( \displaystyle\sum_{i=1}^{10} (i^2 + 3i - 2) \).

View solution

Apply linearity:

\[ \sum_{i=1}^{10}(i^2 + 3i - 2) = \sum_{i=1}^{10} i^2 + 3\sum_{i=1}^{10} i - \sum_{i=1}^{10} 2 \]

Now use the formulas with \(n = 10\):

\( \sum i^2 = \dfrac{10 \cdot 11 \cdot 21}{6} = 385 \)
\( \sum i = \dfrac{10 \cdot 11}{2} = 55 \)
\( \sum 2 = 2\cdot 10 = 20 \)

So the total is

\[ 385 + 3(55) - 20 = 385 + 165 - 20 = 530. \]

The sum equals 530.

Practice Exercises

Exercise 1

Question: Find the first four terms of the sequence \( a_n = n^2 - n \).

View solution

\( a_1 = 1^2 - 1 = 0 \)
\( a_2 = 2^2 - 2 = 2 \)
\( a_3 = 3^2 - 3 = 6 \)
\( a_4 = 4^2 - 4 = 12 \)

Answer: 0, 2, 6, 12.

Exercise 2

Question: Find the first six terms of \( a_1 = -1,\ a_2 = -3,\ a_n = -a_{n-1} - 2a_{n-2} \) for \( n > 2 \).

View solution

\( a_3 = -(-3) - 2(-1) = 3 + 2 = 5 \)
\( a_4 = -(5) - 2(-3) = -5 + 6 = 1 \)
\( a_5 = -(1) - 2(5) = -1 - 10 = -11 \)
\( a_6 = -(-11) - 2(1) = 11 - 2 = 9 \)

Answer: −1, −3, 5, 1, −11, 9.

Exercise 3

Question: Find the sum of the first five terms of \( a_n = n^2 + \dfrac{n+1}{2} \).

View solution

\( a_1 = 1 + 1 = 2 \)
\( a_2 = 4 + 1.5 = 5.5 \)
\( a_3 = 9 + 2 = 11 \)
\( a_4 = 16 + 2.5 = 18.5 \)
\( a_5 = 25 + 3 = 28 \)

Sum: \(2 + 5.5 + 11 + 18.5 + 28 = 65\).

Answer: 65.

Exercise 4 (extra)

Question: Use a power-sum formula to evaluate \( \displaystyle\sum_{i=1}^{n} (2i - 1) \) and verify the result is \(n^2\) for \(n=5\).

View solution

By linearity:

\[ \sum_{i=1}^{n}(2i - 1) = 2\sum_{i=1}^{n} i - n = 2\cdot\frac{n(n+1)}{2} - n = n^2. \]

For \(n=5\): \(1 + 3 + 5 + 7 + 9 = 25 = 5^2\) ✓

Related Practice

Continue with these companion lessons: