4.1 Q-2

Exercise 4.1 — Question 2

Key Concept: Finding Terms of a Sequence

A sequence is a function whose domain is the set of positive integers $\{1,2,3,\dots \}$. If the general (nth) term $a_n$ is given as a formula in $n$, any specific term can be found by substituting the position number for $n$.

Rule: To find the $k^{th}$ term, substitute $n=k$ into the formula $a_n$.

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Worked Examples

Example 1: Given $a_n=\frac{n^2-1}{n^2+1}$, find the first four terms and the $15^{th}$ term.

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

$a_2=\frac{2^2-1}{2^2+1}=\frac{3}{5}$

$a_3=\frac{3^2-1}{3^2+1}=\frac{8}{10}=\frac{4}{5}$

$a_4=\frac{4^2-1}{4^2+1}=\frac{15}{17}$

$a_{15}=\frac{15^2-1}{15^2+1}=\frac{224}{226}=\frac{112}{113}$

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Example 2: Given $a_n=(-1{)}^n\cdot n$, find the first five terms.

$a_1=(-1{)}^1\cdot 1=-1$ $a_2=(-1{)}^2\cdot 2=2$ $a_3=(-1{)}^3\cdot 3=-3$ $a_4=(-1{)}^4\cdot 4=4$ $a_5=(-1{)}^5\cdot 5=-5$

The sequence is $-1,2,-3,4,-5,\dots$ (alternating signs).

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Identifying Arithmetic Sequences

A sequence $a_1,a_2,a_3,\dots$ is arithmetic if there is a constant common difference $d$ such that:

$a_{n+1}-a_n=d\text{for all }n$

The general term of an arithmetic sequence is:

$a_n=a_1+(n-1)d$

Example: Is the sequence $3,7,11,15,\dots$ arithmetic?

$7-3=4,11-7=4,15-11=4$

Yes, it is arithmetic with $a_1=3$ and $d=4$.

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