Definitions and Notation

Theory

A sequence is a function whose domain is the set of positive integers. Each input $n$ produces a term $a_n$, and the terms are written in order. A sequence may be finite or infinite.

Beginner Pattern (Use for Every Problem)

• Identify the starting value of $n$
• Substitute $n = 1, 2, 3, \dots$
• Write the terms in order

Exercises

Exercise 1

Question

Write the first four terms of the infinite sequence

$$\left\{ \frac{1}{3n} \right\}_{n=1}^{\infty}.$$

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Solution

Substitute $n=1,2,3,4$:

$$\frac13,\; \frac16,\; \frac19,\; \frac1{12}.$$

Exercise 2

Question

Write the first four terms of the sequence

$$\left\{ \sin\!\left(\frac{n\pi}{2}\right) - \cos\!\left(\frac{n\pi}{2}\right) \right\}_{n=0}^{\infty}.$$

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Solution

Substitute $n=0,1,2,3$:

$$-1,\; 1,\; 1,\; -1.$$

Exercise 3

Question

Write the first ten terms of the Fibonacci sequence $\{F_n\}$ defined by

$$F_{n+1} = F_n + F_{n-1}, \quad n \ge 2,$$

where $F_1 = 1$ and $F_2 = 1$.

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Solution

Starting from $F_1=1$, $F_2=1$:

$$1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55.$$

Exercise 4

Question

Find the $n$th term (bracket notation) of the sequence

$$2,\; -2,\; 2,\; -2,\; \dots$$

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Solution

The signs alternate and the magnitude is constant:

$$\left\{ 2(-1)^{\,n+1} \right\}_{n=1}^{\infty}.$$

Exercise 5

Question

Find the $n$th term (bracket notation) of the sequence

$$\frac12,\; \frac43,\; \frac94,\; \frac{16}{5},\; \dots$$

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Solution

The numerator is $n^2$ and the denominator is $n+1$:

$$\left\{ \frac{n^2}{n+1} \right\}_{n=1}^{\infty}.$$