4.2 Q-11

Exercise 4.2 — Question 11

Topic: Arithmetic Sequences and Series

Relevant SLO: Solve problems by analyzing arithmetic sequences and series up to $n$ terms.

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Key Concepts

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a fixed constant called the common difference $d$.

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

where:

• $a_n$ = $n^{th}$ term
• $a_1$ = first term
• $d$ = common difference
• $n$ = term number

The common difference is found by: $d=a_2-a_1=a_3-a_2=\cdots =a_n-a_{n-1}$

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Problem

Find the $n^{th}$ term and the indicated term of the arithmetic sequence.

Method:

1. Identify the first term $a_1$.
2. Find the common difference $d=a_2-a_1$.
3. Substitute into $a_n=a_1+(n-1)d$.

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Worked Example

For the arithmetic sequence $3,7,11,15,\dots$, find the $20^{th}$ term.

Step 1: $a_1=3$

Step 2: $d=7-3=4$

Step 3: $a_{20}=3+(20-1)(4)=3+76=79$

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Sum of Arithmetic Series

The sum of the first $n$ terms of an arithmetic series:

$S_n=\frac{n}{2}[2a_1+(n-1)d]=\frac{n}{2}(a_1+a_n)$

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