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.

REVIEW:

arithmetic sequence un = un-1 + dd is the common difference

geometric sequence un = r u∙ n-1

r is the common ratio

Example: 308, 231, 173.25, 128.9375, 97.453125,…

a)Arithmetic or geometric sequence?b)Common ratio?c)Growth or decay? d)By what percentage?e)Write recursive sequence: f)Find 10th term

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 u∙ n-1 Why 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.

Growth Decay

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, …a)What is the rebound ratio for this ball?b)What is the height after the 10th bounce?c)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=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?

Assignment: page 41 1-3, 7-9

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.

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=23,999un=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% = .00625U0=825 Un=(1+.075/12)*un-1

a)What is the balance after 1 month?b)What is the balance after 1 year?c)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