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Introduction to growth & decay

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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 sequenceun = un-1 + d

d is the common difference

geometric sequenceun = r ∙ un-1

r is the common ratio

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

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

U0 = 200Why u0? And not u1?

Un = 1.07 ∙ un-1Why 1.07?

Growth = (100% + P)

Decay = (100% - P)

Important Note: write this as a decimal

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.

GrowthDecay

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?

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?

Example:

U0=1000Un=(1.3)un-1

Growth or decay?By what percentage?

U0=222Un=(0.3) un-1

Growth or decay?By what percentage?

Assignment: page 41 1-3, 7-9

- 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

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

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

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=825Un=(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?

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

Arithmetic or geometric?

What term will give you the first negative number?

Assignment:

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