Arithmetic Sequence Formulas Flashcards

Learning Objectives for Sequences

• Write recursive formulas for both arithmetic and geometric sequences based on specific contexts.

• Write explicit expressions for arithmetic and geometric sequences derived from contextual scenarios.

• Utilize mathematical formulas to determine the value of unknown terms within a sequence.

Writing Formulas for Arithmetic Sequences: Recursive Formulas

• A recursive formula expresses each new term of a sequence based on the preceding term.

• To use a recursive formula, the value of the previous term in the sequence must be known.

• Because they depend on knowing the immediately preceding value, recursive formulas are not the most efficient tool for finding terms far down a sequence, such as a 200th term.

• The general recursive formula for the nth term of an arithmetic sequence is defined as:     $\{a_n = a_{n-1} + d\}$

• In this formula:

  • $a_n$ represents the term being calculated (the nth term).

  • $a_{n-1}$ represents the previous term in the sequence.

  • $d$ represents the common difference, which is the constant value added or subtracted to move from one term to the next.

Application of Recursive Formulas: Negative Sequence Case Study

• Consider the sequence: $-2, -9, -16, -23$.

• Analysis of the sequence shows a constant subtraction of $7$ between terms ($-2$ to $-9$ is $-7$, $-9$ to $-16$ is $-7$, etc.). Therefore, the common difference $d = -7$.

• To find the fifth term ($a_5$) of this sequence, the value of the fourth term ($a_4$) must be utilized.

• Using the notation $a_n = a_{n-1} + d$:

  • $a_5 = a_{5-1} + (-7)$

  • $a_5 = a_4 + (-7)$

• Given that the fourth term ($a_4$) is $-23$, the calculation is:

  • $a_5 = -23 - 7$

  • $a_5 = -30$

• The fifth term in the sequence is $-30$.

Jamal’s Sports Donation: Contextual Arithmetic Sequence

• Jamal owns a sporting goods store and agrees to donate to his old high school’s baseball team equipment fund.

• The donation consists of a flat set amount of $125$ dollars, plus an additional $18$ dollars for every home run the team hits during the season.

• This scenario represents an arithmetic sequence because there is a common difference of $18$ being added for each home run.

• The sequence terms are defined by the number of home runs hit:

  • The first term ($a_1$) is $125$. This represents the set donation when the team has hit zero home runs.

  • Subsequent terms represent the cumulative donation total as more home runs are hit.

Calculating the Eleventh Term in Jamal’s Sequence

• To find the eleventh term ($a_{11}$) using a recursive formula, the tenth term ($a_{10}$) must be known.

• The tenth term ($a_{10}$) in Jamal's sequence has a value of $287$.

• Using the recursive formula for the eleventh term:

  • $a_{11} = a_{11-1} + 18$

  • $a_{11} = a_{10} + 18$

  • $a_{11} = 287 + 18$

• Vertical calculation for accuracy:

  • $287 + 10 = 297$

  • $297 + 8 = 305$

• The eleventh term ($a_{11}$) is $305$.

• Contextual Meaning: In the context of Jamal's donation table, the eleventh term ($a_{11}$) corresponds to exactly $10$ home runs being hit. Therefore, if the team hits $10$ home runs, Jamal will donate a total of $305$ dollars.

Writing Formulas for Arithmetic Sequences: Explicit Formulas

• An explicit formula provides a more efficient way to calculate any term within a sequence because it uses the term's position ($n$) directly.

• Unlike recursive formulas, explicit formulas do not rely on the value of the previous term. This allows for the immediate calculation of high-value terms, such as the 742nd term, without calculating all preceding terms.

• The explicit formula for determining the nth term of an arithmetic sequence is:     $\{a_n = a_1 + d(n - 1)\}$

• In this formula:

  • $a_n$ is the value of the nth term.

  • $a_1$ is the value of the first term in the sequence.

  • $d$ is the common difference.

  • $(n - 1)$ represents the number of common differences added to the first term to reach the nth position.

Application of Explicit Formulas: Jamal’s 35 Home Run Scenario

• To find the total donation for $35$ home runs hit, the position in the sequence must first be determined.

• Since zero home runs is the first term ($a_1$), $35$ home runs corresponds to the 36th term ($a_{36}$).

• Variable values for the formula:

  • $n = 36$

  • $a_1 = 125$

  • $d = 18$

• Substituting into the explicit formula:

  • $a_{36} = 125 + 18(36 - 1)$

  • $a_{36} = 125 + 18(35)$

• The logic holds that there is a flat fee of $125$ dollars plus $18$ dollars multiplied by the $35$ home runs hit.

Application of Explicit Formulas: Jamal’s 48 Home Run Scenario

• To determine the donation amount for $48$ home runs hit, recognize that this represents the 49th term in the sequence ($a_{49}$).

• Substituting into the formula:

  • $a_{n} = a_1 + d(n - 1)$

  • $a_{49} = 125 + 18(49 - 1)$

  • $a_{49} = 125 + 18(48)$

• This calculation adds the initial $125$ dollar donation to the product of $18$ dollars and the $48$ individual home runs hit.

Introduction to Geometric Sequences

• While arithmetic sequences involve a common difference (addition or subtraction), geometric sequences are characterized by a common ratio (multiplication or division).

• Similar to arithmetic sequences, geometric sequences can be described using both recursive and explicit formulas to find specific terms.