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Learning Targets: I can use geometric sequences to model growth and decay. I can use recursion notation to model growth and decay. Introduction to growth & decay. REVIEW: arithmetic sequence u n = u n-1 + d d is the common difference geometric sequence u n = r ∙ u n-1

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introduction to growth decay
Learning Targets:
    • I can use geometric sequences to model growth and decay.
    • I can use recursion notation to model growth and decay.

Introduction to growth & decay

slide2
REVIEW:

arithmetic sequenceun = un-1 + d

d is the common difference

geometric sequenceun = r ∙ un-1

r is the common ratio

slide3
Example:

308, 231, 173.25, 128.9375, 97.453125,…

Arithmetic or geometric sequence?

Common ratio?

Growth or decay?

By what percentage?

Write recursive sequence:

Find 10th term

slide4
Example: Jim deposits $200 in an account which gets 7% interest. Write a recursive sequence.

U0 = 200 Why u0? And not u1?

Un = 1.07 ∙ un-1 Why 1.07?

slide5
Growth = (100% + P)

Decay = (100% - P)

Important Note: write this as a decimal

slide6
What do the graphs of growth and decay look like??

The same amount is NOT being added or subtracted, so the graph is NOT linear.

Growth Decay

slide7
Example: Suppose the initial height from which a rubber ball drops is 100 cm. The rebound heights to the nearest cm are 80, 64, 51, 41, …

What is the rebound ratio for this ball?

What is the height after the 10th bounce?

After how many bounces will the ball be less than 1 cm?

slide8
Example: A book store is going out of business. They will mark their books down an additional 10% each week until they sell all of their inventory. If a book costs $35, how much will it cost after 4 weeks?
slide9
Example:

U0=1000 Un=(1.3)un-1

Growth or decay? By what percentage?

U0=222 Un=(0.3) un-1

Growth or decay? By what percentage?

more examples of growth decay
Learning Targets:
    • I can use geometric sequences to model growth and decay.
    • I can use recursion notation to model growth and decay.

More examples of Growth & Decay

slide12
Example: An automobile depreciates as it gets older. Suppose that a particular automobile loses 1/5 of its value each year. Write a recursive formula to find the value of this car when it is 6 years old if it costs $23,999 when it was new.

u0=23,999

un=4/5(un-1) or un=.8*un-1

answer: after 6 years it is worth $6291.19

slide13
Carbon dating is used to find the age of ancient remains of once-living things. Carbon-14 is found naturally in all living things, and it decays slowly after death. About 11.45% of it decays every 1,000 years. Let 100% or 1 be the beginning amount of Carbon-14. At what point will less than 8% remain? Write a recursive formula you used.

Answer:

U0=1 Un=(1-.1145)un-1(or use .8855 in parenthesis)

22,00 years

slide14
Suppose $825 is deposited in an account that earns 7.5% annual interest and no more deposits or withdrawals are made. If the interest is compounded monthly, what is the monthly rate?

(7.5%/12 months) = 0.625% = .00625

U0=825 Un=(1+.075/12)*un-1

What is the balance after 1 month?

What is the balance after 1 year?

What is the balance after 35 months?

slide15
Write a recursive formula for 115, 103, 91, 79,…

Arithmetic or geometric?

What term will give you the first negative number?

slide16
Assignment:

page 41 10, 11, 12abc, 13, 18

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