Do Now – Complete the volume Do Now – Complete the volume problems in terms of pi problems in terms of pi

NO CALCULATORSNO CALCULATORS

EXAMPLEEXAMPLEFind the volume of a cone, in terms of pi, with Find the volume of a cone, in terms of pi, with a radius of 3 inches and a height of 6 inches. a radius of 3 inches and a height of 6 inches.

V = 1/3V = 1/3π(3π(322)(6))(6)

V = 1/3(9)(6) πV = 1/3(9)(6) π

V = 3(6) πV = 3(6) π

V = 18πV = 18π

10-6-1410-6-14

Recursive FormulasRecursive FormulasFor Arithmetic For Arithmetic

SequencesSequences

Do you know the definition Do you know the definition of “recursive”?of “recursive”?

According to Merriam Webster:According to Merriam Webster:

relating to a procedure that can relating to a procedure that can repeat itself indefinitelyrepeat itself indefinitely

04/19/2304/19/23

10-6-1410-6-14

Mathematically SpeakingMathematically SpeakingRecursive Formula:Recursive Formula:–Formula where each term is based on the term Formula where each term is based on the term before it.before it.

Arithmetic Sequence:Arithmetic Sequence:–Sequence with a constant difference between Sequence with a constant difference between terms.terms.

Domain of arithmetic sequencesDomain of arithmetic sequences

You can use arithmetic sequences You can use arithmetic sequences as linear functions to model real-as linear functions to model real-world situations. While the domain world situations. While the domain of some linear functions is the set of of some linear functions is the set of all real numbers, the domain of an all real numbers, the domain of an arithmetic sequence as a linear arithmetic sequence as a linear function is the set of counting function is the set of counting numbers, {1, 2, 3, 4, …}.numbers, {1, 2, 3, 4, …}.

04/19/2304/19/23

ExampleExample• Consider the sequence generated byConsider the sequence generated by

2000, 2040, 2080, 2120, 2160,. . .2000, 2040, 2080, 2120, 2160,. . .

• Describe this sequence in words.Describe this sequence in words.

n = positive integers ≥ 2

How is a recursive How is a recursive function represented?function represented?

1-6-141-6-14

10/6/1410/6/14YOU TRY: Write an arithmetic sequence for YOU TRY: Write an arithmetic sequence for

the range of this sequence, the range of this sequence,

If you buy a new car, you might be advised If you buy a new car, you might be advised to have an oil change after driving 1000 to have an oil change after driving 1000 miles and every 3000 miles thereafter. Then miles and every 3000 miles thereafter. Then the following sequence gives the mileage the following sequence gives the mileage when oil changes are required:when oil changes are required:

1000 4000 7000 10000 13000 160001000 4000 7000 10000 13000 16000

n = positive integers ≥ 2

Another example; This time write the Another example; This time write the recursive formularecursive formula

• Briana borrowed $870 from her parents for Briana borrowed $870 from her parents for airfare to Europe. She will pay them back airfare to Europe. She will pay them back at the rate of $60.00 per month. Let aat the rate of $60.00 per month. Let ann be be

the amount she still owes after n months. the amount she still owes after n months. Find a recursive formula for this sequence. Find a recursive formula for this sequence.

n = positive integers ≥ 2

Graph of an Arith. Seq.Graph of an Arith. Seq.• Discrete Domain and Discrete Domain and

RangeRange

• Constant Increase orConstant Increase or

DecreaseDecrease

• Your first input is 1, Your first input is 1, not 0not 0

04/19/2304/19/23

3-83-8Explicit FormulasExplicit Formulas

For ArithmeticFor ArithmeticSequencesSequences

04/19/2304/19/23

Arithmetic SequencesArithmetic SequencesExplicit FormulaExplicit Formula

– Formula where any term can be found by Formula where any term can be found by substituting the number of that term.substituting the number of that term.

– We can develop an explicit formula for an We can develop an explicit formula for an Arithmetic Sequence from the recursive Arithmetic Sequence from the recursive formulaformula

04/19/2304/19/23

Explicit FormulaExplicit Formula

nn

11

22

33

44

aann

10001000

1000+3000=40001000+3000=4000

4000+3000=70004000+3000=7000

7000+3000=100007000+3000=10000

dd

30003000

30003000

30003000

2;3000

1000

1

1

naa

a

nn

dnaan )1(1

04/19/2304/19/23

• So, for our oil change example, the explicit So, for our oil change example, the explicit formula looks like:formula looks like:

3000)1(1000 nan

10/22/1310/22/13

ExamplesExamples1.1. Find the 40Find the 40thth term of the arithmetic term of the arithmetic

sequence 100,97,94,91,…..sequence 100,97,94,91,…..

2.2. In a concert hall the 1In a concert hall the 1stst row has 20 seats row has 20 seats in it, and each subsequent row has 2 in it, and each subsequent row has 2 more seats than the row in front of it. If more seats than the row in front of it. If the last row has 64 seats, how many the last row has 64 seats, how many rows are in the concert hall?rows are in the concert hall?

3.3. Answer 23 rows!Answer 23 rows!