Example of geometric progressions;
a). 2, 4, 8, 16, 32, ...
b) 1, 2/3 , 4/9, 8/27, ...
c) a, ar, ar2, ar3, ...
You may recall that in a geometric progression we multiply each term by some fixed number to get the next term.
The formula for the sum of n terms of the geometric progression is
(1.1)That geometric series has a sum if and only if |r| < 1, and in this case the sum is
The series is then called convergent.
(1.2)Problem, Section 1
1. In the bouncing ball example above, find the height of the tenth rebound, and the distance traveled by the ball after it touches the ground the tenth time. Compare this distance with the total distance traveled.(Suppose a bouncing ball rises each time to 2/3 of the height of the previous bounce then it would represent the heights of successive bounces in yards if the ball is originally dropped from a height of 1 yd).
Answer:a. The height of the tenth rebound
1, 2/3 , 4/9, 8/27, ...
U1 = a = 1 (the originally height)
U2 = 2/3(the height of the first rebound)
U3 = 4/9(the height of the second rebound)
'
'
U11 = ? (the height of the eleventh rebound)
Un= a.rn-1
U11= 1.(2/3)11-1
U11= 1.(2/3)10
U11= 0.017 yd
So, the height of the tenth rebound is 0.017 yd.
b. The distance traveled by the ball after it touches the ground the tenth time.
c. The total distance traveled.
2. Derive the formula (1.1) for the sum Sn of the geometric progression Sn = a+ar+ar2 + · · · + arn-1. Hint: Multiply Sn by r and subtract the result from Sn; then solve for Sn. Show that the geometric series converges if and only if |r| < 1; also show that if |r| < 1, the sum is given by equation (1.2).
Use equation (1.2) to find the fractions that are equivalent to the following repeating decimals:
3. 0.55555 · · ·
4. 0.818181 · · ·
5. 0.583333 · · ·
6. 0.61111 · · ·
7. 0.185185 · · ·
8. 0.694444 · · ·
9. 0.857142857142 · · ·
10. 0.576923076923076923 · · ·
11. 0.678571428571428571 · · ·
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