In Example 3(b) on page 467, we found partial sums for the geometric series with a = 250

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula. 3 +3 2 + 3 22

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula. 50 + 50(0.9) + 50(0.9)2 + 50(0.9)3

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. 5 10 + 20 40 + 80

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. 1 + 1 2 + 1 3 + 1 4 + 1 5 +

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. 2 + 1 + 1 2 + 1 4 + 1 8 +

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. 1 1 2 + 1 4 1 8 + 1 16 +

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. 1 + 2z + (2z)2 + (2z)3 +

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. 1 + x+ 2x2 + 3x3 + 4x4 +

In Problems 39, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. y2 + y3 + y4 + y5 +

In Problems 1021, find the sum, if it exists. 5 + 5 3 + 5 32 + + 5 312

In Problems 1021, find the sum, if it exists. 100 + 100(0.85) + 100(0.85)2 + + 100(0.85)10

In Problems 1021, find the sum, if it exists. 1000 + 1000(1.05) + 1000(1.05)2 +

In Problems 1021, find the sum, if it exists. 75 + 75(0.22) + 75(0.22)2 +

In Problems 1021, find the sum, if it exists. 20 + 20(1.45) + 20(1.45)2 + + 20(1.45)14

In Problems 1021, find the sum, if it exists. 500(0.4) + 500(0.4)2 + 500(0.4)3 +

In Problems 1021, find the sum, if it exists. 31500 + 6300 + 1260 + 252 +

In Problems 1021, find the sum, if it exists. 3 + 3 2 + 3 4 + 3 8 + + 3 210

In Problems 1021, find the sum, if it exists. 1000 + 1500 + 2250 + 3375 + 5062.5 +

In Problems 1021, find the sum, if it exists. 200 + 100 + 50 + 25 + 12.5 +

In Problems 1021, find the sum, if it exists. 65 + 65 1.02 + 65 (1.02)2 + + 65 (1.02)18

In Problems 1021, find the sum, if it exists. 2 + 1 1 2 + 1 4 1 8 + 1 16

In Example 3(b) on page 467, we found partial sums for the geometric series with a = 250 and r = 1.2. Find the partial sums Sn for n = 5, 10, 15, 20. As n gets larger, do the partial sums appear to grow without bound, as expected if r > 1?

In Example 3(a), we found partial sums of the geometric series with a = 10 and r = 0.75 and showed that the sum of this series is 40. Find the partial sums Sn for n = 5, 10, 15, 20. As n gets larger, do the partial sums appear to be approaching 40?

A repeating decimal can always be expressed as a fraction. This problem shows how writing a repeating decimal as a geometric series enables you to find the fraction. (a) Write the repeating decimal 0.232323. . . as a geometric series using the fact that 0.232323 . . . = 0.23 + 0.0023 + 0.000023 + . (b) Use the formula for the sum of a geometric series to show that 0.232323 . . . = 23/99.

Every month, $500 is deposited into an account earning 0.1% interest amonth, compounded monthly. (a) How much is in the account right after the 6th deposit? Right before the 6th deposit? (b) How much is in the account right after the 12th deposit? Right before the 12th deposit?

Each year, a family deposits $5000 into an account paying 1.25% interest per year, compounded annually. How much is in the account right after the 25th deposit?

A smoker inhales 0.4 mg of nicotine from a cigarette. After one hour, 71% of the nicotine remains in the body. If a person smokes one cigarette every hour beginning at 7 am, how much nicotine is in the body right after the 11 pm cigarette?

Each morning, a patient receives a 25 mg injection of an anti-inflammatory drug, and 40% of the drug remains in the body after 24 hours. Find the quantity in the body: (a) Right after the 3rd injection. (b) Right after the 6th injection. (c) In the long run, right after an injection.

In Example 4 on page 468, we saw that if 50 mg of quinine is given every 24 hours, the long-run quantity of quinine in the body is about 65 mg right after a dose and about 15 mg right before a dose. The concentration of quinine in the body is measured in milligrams of quinine per kilogram of body weight. To be effective, the average concentration of quinine in the body must be at least 0.4 mg/kg. Concentrations above 3.0 mg/kg are not safe. (a) Estimate the average quantity of quinine in the body in the long run by averaging the long-run quantities of quinine in the body right after a dose and right before a dose. (b) Find the average concentration for a person weighing 70 kilograms. Is this treatment safe and effective for such a person? (c) For what range of weights would this treatment produce a long-run average concentration that is (i) Too low? (ii) Unsafe?

A ball is dropped from a height of 10 feet and bounces. Each bounce is 3 4 of the height of the bounce before. Thus, after the ball hits the floor for the first time, the ball rises to a height of 10( 3 4) = 7.5 feet, and after it hits the floor for the second time, it rises to a height of 7.5( 3 4) = 10(3 4 )2 = 5.625 feet. (a) Find an expression for the height to which the ball rises after it hits the floor for the nth time. (b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times. (c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the nth time. Express your answer in closed form.