Arithmetic Sequences and Series

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Topic 3 Lesson 4

Arithmetic Sequences

  • The number of seats in each row of an auditorium can be observed in the sequence: 3, 7, 11, 15, 19…

    • **Explanation of the Sequence:

      • The first row has 3 seats.

      • The second row has 7 seats.

      • The increments continue in a consistent manner.**

Sequence Completion

a) Next Two Numbers

  • To find the next two numbers in the sequence 3, 7, 11, 15, 19:

    • The next numbers are calculated by adding the common difference, which is 4:

    • After 19, the next term is 19 + 4 = 23.

    • Then, 23 + 4 = 27.

    • So, the complete sequence is: 3, 7, 11, 15, 19, 23, 27…

b) Explanation of Finding the Next Number

  • The pattern of the sequence is identified by observing the constant difference:

    • Each term increases by 4.

    • This constant increase allows for predictable continuation of the sequence.

c) Equation to Model This Pattern

  • Explicit Formula for Arithmetic Sequence:

    • The general formula for the nth term, denoted as $an$, is given by: a</em>n=a1+(n−1)imesda</em>n = a_1 + (n-1) imes d
      where,

    • $a_1$ = first term

    • $d$ = common difference

Important Concepts Definition

  • Definition of a Sequence:

    • A sequence is an ordered list of numbers that often forms a pattern. Each identifiable number in the list is a term of the sequence.

  • Subscript Notation:

    • The nth term of a sequence is denoted as $a_n$.

    • Thus, $a1$ refers to the first term, $a2$ to the second term, etc.

Definition of Key Terminology

  • Arithmetic Sequence: An arithmetic sequence is one in which the difference between any two consecutive terms is constant.

  • Common Difference (d): The constant difference between consecutive terms in an arithmetic sequence.

    • For the sequence 3, 7, 11, 15, 19, the common difference is $d = 4$.

Essential Questions

  • How are arithmetic sequences related to linear functions?

    • Arithmetic sequences can be graphically represented as linear functions where the y-intercept is the first term of the sequence and the slope corresponds to the common difference.

Example Exercises

Example 1 - Explicit Formula

a) Explicit Formulation for Steps of Pyramid

  • The height of the nth step of a pyramid follows an arithmetic pattern:

    • First step: 30 cm off the ground.

    • Heights of subsequent steps: each is 26 cm tall.

    • Find the nth term in the sequence of heights for steps: an=30+(n−1)imes26a_n = 30 + (n-1) imes 26

Example 2 - Application of Arithmetic Sequences

  • A high school auditorium example:

    • 18 seats in the first row, 26 seats in the fifth row.
      A. Find the explicit definition for the sequence.
      B. Determine how many seats are in the 12th row.

    • Identified Explicit Formula:
      a<em>n=a</em>1+(n−1)imesda<em>n = a</em>1 + (n - 1) imes d

Finding the Sum of an Arithmetic Sequence

  • To find the sum of the terms in an arithmetic sequence:

    • Example Sequence: 1, 4, 7, 10, 13

    • General Formula for an arithmetic series:
      S<em>n=racn2(a</em>1+an)S<em>n = rac{n}{2} (a</em>1 + a_n)
      where

    • $S_n$ = sum of the first n terms

    • $n$ = total number of terms

Example 3

  • What is the sum of the sequence 2, 6, 10, 14, 18, 22?

    • Determine how many terms to include and apply the summation formula.