Algebra II Honors/Gifted
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Assignment #______
SEQUENCES and SERIES MODELS GRADED PROBLEMS
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1) Chewing Gum Problem. Phoebe Small get caught chewing gum in algebra class. Quickly,
she explains to her teacher that she is conducting an experiment in geometric sequences and
series, since with each chew she gets 0.9 times as much flavor as she did with the preceding
chew.
a) If she gets 40 “squirts” of flavor with the first chew,
how many squirts of flavor will she get with the tenth chew?
b) What total amount of flavor will she get in the first ten chews?
c) Phoebe’s teacher tells her that her punishment is to chew the gum
forever . What total amount of flavor will she receive?
2) Bouncing Ball Problem. Calvin Butterball drops a superball from his bedroom window 20
meters above the ground. The ball bounces back up to 90% of its previous height with each
successive bounce.
a) How far does the ball travel vertically between the 1st and 2nd bounces?
b) Show that the distance the ball travels vertically between bounces are terms of a
geometric sequence.
c) Find the number of meters the ball travels vertically between the 6th and 7th bounces.
d) Find the total vertical distance in meters the ball travels between the 1st and 7th
bounces.
e) If the ball continues to bounce in this manner until it comes to rest, how far will the
ball have travelled vertically from the time it was dropped from the window?
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3) Calvin and Phoebe Problem. Calvin Butterball and Phoebe Small start walking at the same
time from the same point and in the same direction. Phoebe starts with a 12 inch step and
increases her stride by ½ inch each step. She goes 21 steps, then stops. Calvin starts with a 36
inch step and each subsequent step is 90% as long as the previous step.
a) What kind of sequence do Phoebe’s steps follow? Explain.
b) What kind of sequence do Calvin’s steps follow? Explain.
c) How long is Phoebe’s last step?
d) How long is Calvin’s 21st step?
e) After each has taken 21 steps, who is ahead? By how many inches?
f) If Calvin keeps on walking in the same manner forever, will he ever catch up to
Phoebe? Explain.
4) Lion Hunting in Africa. When asked to go about catching
a lion in Africa, a mathematician replied, “That’s easy! You just
build a fence across Africa, dividing it in half. Then you find which
side of the fence the lion is on, and build another fence, dividing that
half in half. You continue the process until the lion is in a corral.”
Africa has an area of 30,300,000 square kilometers. How many fences
would you have to build to trap the lion in a corral less than ten square
meters in area?
5) Interest Compounding Problem. Suppose that Leon Federal Savings pays 6.25% per year
interest, compounded annually and the Lewis State Bank pays 6% per year interest compounded
monthly. Phoebe has $1000 to invest and wants to know which bank has the better deal.
Assume no further deposits nor withdrawals.
a) How much could Phoebe withdraw from each bank at the end of ten full years?
On this basis, which seems better, a higher interest rate or more frequent compounding?
b) How much could Phoebe withdraw from each account at the end of 9 years, 11
months? Remember, if you withdraw before the end of a compounding period, you get
no interest for that period. Which type of account would be better if you are likely to
have to withdraw your money on short notice?
c) What annual interest rate, compounded annually, would you need in order to make the
same amount of money at the end of ten years as you would make with 6% compounded
monthly?
ANSWERS
1)a) about 15.5 squirts
b) about 260.5 squirts
c) 400 squirts
2)a) 36 m
b) r = 0.9
c) about 21.26 m
d) about 168.68 m
e) 380 m
3)a) arithmetic, d = 1/2
b) geometric, r = 0.9
c) 22 in
d) about 4.38 in
e) Phoebe, by about 36 in
f) Yes. Calvin converges to 360 in
4) 42
5)a) $1833.54; $1819.40; higher interest rate
b) $1725.68; $1810.35; frequent compounding
c) 6.16778%