Arithmetic Sequences and Formulas

Formulas for Arithmetic Sequences

Learning Outcomes

  • Write an explicit formula for an arithmetic sequence.

  • Write a recursive formula for the arithmetic sequence.

Using Explicit Formulas for Arithmetic Sequences

  • An arithmetic sequence can be seen as a function on the domain of natural numbers and it is a linear function with a constant rate of change.

  • The common difference is the constant rate of change (slope of the function).

  • A linear function can be constructed if we know the slope and the vertical intercept.

  • Explicit Formula: an = a1 + d(n - 1)

    • Where:

      • a_n is the nth term.

      • a_1 is the first term.

      • d is the common difference.

      • n is the term number.

  • The y-intercept can be found by subtracting the common difference from the first term.

Example

Consider a sequence with a common difference of -50.

  • It represents a linear function with a slope of -50.

  • To find the y-intercept:

    • Subtract -50 from 200.

    • 200 - (-50) = 200 + 50 = 250

  • Alternative method to find the y-intercept: graph the function and see where a line connecting the points intersects the vertical axis.

  • Slope-intercept form of a line: y = mx + b

    • In sequences, a_n replaces y and n replaces x.

  • If slope and y-intercept are known, they can be substituted for m and b in y = mx + b.

    • With a slope of -50 and a y-intercept of 250, the equation is: a_n = -50n + 250

Sequences as Linear Functions

  • If n represents the input and an^{} represents the output, an arithmetic sequence can be visualized as a linear function of the form y = mx + b or an=dn+a_0

  • Each point on the graph is of the form (n, a_n), and the common difference is the slope of the line.

  • If n = 0 in the explicit form an = a1 + d(n - 1), then a0 = a1 - d.

    • The vertical intercept a_0 can be found by subtracting the common difference from the first term.

  • An alternative explicit formula for the above example is: an = 200 - 50(n - 1), which simplifies to an = -50n + 250.

A General Note

Explicit Formula for an Arithmetic Sequence:

  • an = a1 + d(n - 1)

How To: Write an Explicit Formula

Given the first several terms:

  1. Find the common difference, d.

  2. Substitute the common difference and the first term into an = a1 + d(n - 1).

Example: Writing the nth Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence: \left{ 2, 12, 22, 32, 42, … \right}

Solution

  • Find the common difference:

    • d = a2 - a1 = 12 - 2 = 10

  • Substitute the common difference and the first term into the formula:

    • a_n = 2 + 10(n - 1)

    • Simplify:

      • a_n = 10n - 8

  • Analysis: The graph of this sequence has a slope of 10 and a vertical intercept of -8.

Recursive Formula for Arithmetic Sequences

  • Some arithmetic sequences are defined in terms of the previous term using a recursive formula.

  • A recursive formula allows finding any term using the preceding term; each term is the sum of the previous term and the common difference (d).

  • As with any recursive formula, the first term must be given.

  • an = a{n-1} + d, n \geq 2

A General Note

Recursive Formula for an Arithmetic Sequence:

  • an = a{n-1} + d, n \geq 2

How To: Write a Recursive Formula

Given an arithmetic sequence:

  1. Subtract any term from the subsequent term to find the common difference.

  2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.

  • Either the explicit or recursive form may describe an arithmetic sequence when the first term is known.

Example: Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence: \left{ -18, -7, 4, 15, 26, … \right}

Solution

  • The first term is -18.

  • Find the common difference:

    • d = -7 - (-18) = 11

  • Substitute the initial term and the common difference into the recursive formula:

    • a_1 = -18

    • an = a{n-1} + 11, for n \geq 2

  • Analysis: The common difference is the slope of the line formed when graphing the terms. The growth pattern shows a constant difference of 11 units.

How To: Common Difference Subtraction

  • You can subtract any term in the sequence from the subsequent term to find the common difference, but subtracting the first term from the second term is often the easiest.

Find the Number of Terms in an Arithmetic Sequence

  • Explicit formulas determine the number of terms in a finite arithmetic sequence.

  • We need to find the common difference and determine how many times to add the common difference to the first term to obtain the final term.

How To: Find the Number of Terms

Given the first three terms and the last term:

  1. Find the common difference, d.

  2. Substitute the common difference and the first term into an = a1 + d(n - 1).

  • Given the explicit form of an arithmetic sequence, an = a1 + d(n - 1), if known values are substituted for all but one component, we can solve for the missing one.

  • In this case, a certain finite sequence is arithmetic, and we know the first term a1 and the final term an.

  • We can calculate the common difference from any two consecutive terms.

  • Substituting these known values into the explicit formula allows solving for the number of terms n without generating them.

Example: Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence: \left{ 8, 1, -6, …, -41 \right}

Solution

  • Find the common difference:

    • 1 - 8 = -7

  • The common difference is -7.

  • Substitute the common difference and the initial term into the n^{th} term formula:

    • an = a1 + d(n - 1)

    • a_n = 8 + -7(n - 1)

    • a_n = 15 - 7n

  • Substitute -41 for a_n and solve for n:

    • -41 = 15 - 7n

    • 8 = n

  • There are eight terms in the sequence.

Solving Application Problems with Arithmetic Sequences

  • In many application problems, using an initial term of a0 instead of a1 makes sense.

  • The explicit formula is altered slightly to account for the difference in initial terms:

    • an = a0 + dn

  • An arithmetic sequence can be represented as a linear function with input n, output a_n, and common difference (or slope) of d.

Example: Solving Application Problems with Arithmetic Sequences

A five-year-old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.

  1. Write a formula for the child’s weekly allowance in a given year.

  2. What will the child’s allowance be when he is 16 years old?

Solution

  1. The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.

    • Let A_n be the amount of the allowance and n be the number of years after age 5.

    • Using the altered explicit formula for an arithmetic sequence, we get:

      • A_n = 1 + 2n

  2. Find the number of years since age 5 by subtracting:

    • 16 - 5 = 11

  3. We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.

    • A_{11} = 1 + 2(11) = 23

  4. The child’s allowance at age 16 will be $23 per week.