Exercise 4.1 — Q1: Sequences (General Term & Domain)

Key Concepts

A sequence is a function whose domain is the set of positive integers $\{1,2,3,\dots \}$. Each positive integer $n$ is the index (position) of a term, and the output $a_n$ is the $n^{th}$ term of the sequence.

General Term

If the general term $a_n$ is given as a formula in $n$, then:

• The 1st term is found by substituting $n=1$
• The 2nd term is found by substituting $n=2$
• The $k^{th}$ term is found by substituting $n=k$

Example: If $a_n=\frac{n^2-1}{n^2+1}$, find the first four terms.

$a_1=\frac{1-1}{1+1}=\frac{0}{2}=0$

$a_2=\frac{4-1}{4+1}=\frac{3}{5}$

$a_3=\frac{9-1}{9+1}=\frac{8}{10}=\frac{4}{5}$

$a_4=\frac{16-1}{16+1}=\frac{15}{17}$

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Identifying Arithmetic and Geometric Sequences

Arithmetic Sequence

A sequence $a_1,a_2,a_3,\dots$ is arithmetic if there exists a constant $d$ (common difference) such that: $a_{n+1}-a_n=d\text{for all }n$ The general term is: $a_n=a_1+(n-1)d$

Geometric Sequence

A sequence $a_1,a_2,a_3,\dots$ is geometric if there exists a constant $r$ (common ratio) such that: $\frac{a_{n+1}}{a_n}=r\text{for all }n$ The general term is: $a_n=a_1\cdot r^{n-1}$

Harmonic Sequence

A sequence is harmonic if the reciprocals of its terms form an arithmetic sequence. That is, $a_1,a_2,a_3,\dots$ is harmonic if $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots$ is arithmetic.

Example: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ is harmonic because $1,2,3,4,\dots$ is arithmetic.

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Worked Problems

Q1(a): Write the first five terms of the sequence $a_n=2n-1$.

$a_1=1,a_2=3,a_3=5,a_4=7,a_5=9$

This is an arithmetic sequence with $a_1=1$ and $d=2$.

Q1(b): Write the first four terms of $a_n=3\cdot 2^{n-1}$.

$a_1=3,a_2=6,a_3=12,a_4=24$

This is a geometric sequence with $a_1=3$ and $r=2$.