4.1 Q-3

Exercise 4.1 — Question 3

Objective

Given the general term (formula) of a sequence, find specific terms and determine whether the sequence is arithmetic.

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Key Concept: Evaluating Terms from a General Formula

If the $n^{th}$ term of a sequence is given by a formula $a_n$, then the $k^{th}$ term is found by substituting $n=k$ into the formula.

$a_k=a_n{|}_{n=k}$

The domain of a sequence is the set of positive integers $\{1,2,3,\dots \}$.

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Worked Example

Given: $a_n=\frac{n^2-1}{n^2+1}$

Find: $a_1,a_2,a_3,a_4$ and determine if the sequence is arithmetic.

Solution:

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

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

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

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

Check for arithmetic sequence:

A sequence is arithmetic if the difference between consecutive terms is constant: $d=a_{n+1}-a_n=\text{constant}$

$a_2-a_1=\frac{3}{5}-0=\frac{3}{5}$

$a_3-a_2=\frac{4}{5}-\frac{3}{5}=\frac{1}{5}$

Since $\frac{3}{5}\ne \frac{1}{5}$, the common difference is not constant.

$\therefore \text{The sequence is NOT arithmetic.}$

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Identifying an Arithmetic Sequence

A sequence $a_1,a_2,a_3,\dots$ is arithmetic if and only if: $a_{n+1}-a_n=d(\text{constant for all }n)$

The general term of an arithmetic sequence is: $a_n=a_1+(n-1)d$

where $a_1$ is the first term and $d$ is the common difference.

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