Math 12 Pre-calculus

Infinite geometric series

If a geometric sequence has a ratio that falls between -1 and 1 (and is not zero), we noticed that the nth term, as n becomes large, approaches zero. i.e.

We can conclude that exists and is calculable because of the fact that

Thus, It is possible to sum an infinite number of terms in a geometric series if the ratio falls between -1 and 1 (and is not zero).

Thus, if and

1. Determine if each series exists. If it does not, explain why. If it does, evaluate it, expressing the answer in simplest exact form:

a) 5 + 7 + 9 + 11 + …. b) c) 7 + 14 + 28 + 56 +…

d) e) f)

2. A ball is dropped and left to bounce repeatedly. It takes 1 second to fall and hit the ground. As it climbs, it requires 90% of this one second to reach its new maximum height. If the ball continues bouncing in this pattern (90% of the previous time), how long does it take for the it to come to rest?

3. Express each as a fraction by using an infinite series:

a)

b)

c) 0.9999999999…..

d) Wait, what? Does your answer to part C make sense?

4. The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second square.

(a) (i) Show that PQ = cm.

(ii) Find the area of PQRS.

b) The midpoints (W, X, Y, Z) of the sides of PQRS are now joined to form a third square, as shown.

(i) Write down the area of the third square, WXYZ.

(ii) Show that the areas of ABCD, PQRS, and WXYZ form a geometric
sequence. Find the common ratio of this sequence.

c) The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. Calculate the sum of the areas of all the squares.

5. Tax rebates and the multiplier effect. Economic theory suggests that issuing a tax rebate to a consumer can have an effect on the economy that is much more powerful than the actual rebate. Consider: The government issues a tax rebate of $1000 to everyday people. The government also assumes that a consumer spends 70% of his/her $1000 rebate ($700). Then, the business or businesses that received this $700 will spend 70% of it (70% of $700 = $490). In turn, each new business(es) then spend 70% of this $490 they received. This process continues indefinitely, such that the total amount of money spent from this rebate is $700 + $490 + $343 + …..

a) how much money from this $1000 rebate is eventually spent and pushed back into the economy?

b) Instead, the government issues a $5000 tax rebate and assumes that consumers and businesses spend 80% of new income. How much money is sent back to the local economy from this individual tax rebate?

c) From part b) If the local sales tax (like GST or HST) is 12%, how much does this tax rebate actually end up costing the government?

Answers:

Doesn’t exist (arithmetic) doesn’t exist (geometric, but r > 1)

6 not geometric (but terms are getting smaller… does series exist? )

19 seconds 79/90 508/99 1 (what?)

8, 4, r = ½ , 32 $2333.33 $20,000

Govnt collects $2400 in tax per rebate, so net cost to government is $2600.