EXERCISE 136

1. Find the 8th term of the series 5, 10, 20, ........ 2. Find the 8th term of the series 2, 6, 18, ........

3. Find the 7th term of the series 3, 6, +12,....

4. Find the 6th term of the series 4, -6, +9, .... 5. Find the 10th term of the series 81, 27, 9,.... 6. Find the 7th term of the series

- 4, + §, — 16, ...

...

to 7 terms.

14. a9+ ab + a2b2, ·

15. a9 — ab+a7b2, ... to 7 terms.

16. Find the geometric mean between 2 and 50.

17. Prove that the geometric mean of two numbers is equal

to the mean proportional between the numbers.

18. Find the geometric mean between a2 — b2 and a+b

19. Insert 3 geometric means between 4 and 324.
20. Insert 3 geometric means between 3 and 48.
21. Given r=4, n = 3, 1 = 80, find a and s.
22. Given r = 3, n = 3, l = 18, find a and s.
23. Given r=4, n = 3, .s = 105, find a and 7.
24. Given r= 5, n = 4, s = 780, find a and ¿.

a b

25. Given a = 2, r = 4, 7 = 32, find n and s.
26. Given a = 5, r = 4, 180, find n and s.
27. Given a = 4, n = 3, l = 64, find r and s.
28. Given a = 5, n=3, l = 125, find r and s.
29. Given a = 15, r = 3, s= 600, find n and 7.
30. Given a = 15, r = 4, s=5115, find n and 1.
31. Find s in terms of a, r, and 7.

32. Find a in terms of r, n, and 1.

33. Find a in terms of r, n, and s.

34. Find r in terms of a, n, and 7.

35. Find the sum of the series √3, 3, 3 √3, ... to 5 terms. 36. Find the sum of the series 1, 2, 4, 8, to n terms.

37. The fourth term of a G. P. is 135, the seventh term 3645. Find the series.

38. The sum of the third and fifth terms of a G. P. is 90, and the sum of the sixth and eighth terms is 2430. Find the series.

39. The population of a city is 100,000, and it increases 50% every 4 years. What will the population be in 20 years?

40. A sum of money invested at 6% compound interest doubles itself in 12 years. What will $1.00 invested at 6% compound interest amount to in 240 years?

of

INFINITE GEOMETRIC PROGRESSION

367. If the value of r of a G. P. is less than unity, the value decreases, if n increases. The formula for the sum may

By taking n sufficiently large, r", and hence

made less than any assignable number.

Consequently, by taking a sufficiently large number of terms,

a

s can be made to differ from by less than any given num

1

r

ber, however small. This is usually expressed by the formula

Ex. 1. Find the sum to infinity of the series 1, − 1, 1, ·

12. The sum of an infinite G. P. is 6, and the first term is

4. Find the series.

13. A ball is thrown vertically upwards to a height of 16 feet. After striking the ground it rebounds to three fourths the height it dropped from, and so on for each successive rebound. What is the entire distance traveled by the ball until it comes to rest?

14. Under the conditions given in Ex. 13, the time the ball needs for the first ascension and fall is 2 seconds, and the time between any two successive rebounds is equal to the preceding similar period multiplied by V3. In what time does the ball come to rest?

CHAPTER XXII

BINOMIAL THEOREM

PROOF BY MATHEMATICAL INDUCTION

368. The following example explains the demonstration by mathematical induction, a method frequently used in algebra. To prove that

13+23+33... n3: =

n2 (n + 1)2.

4

But (2) expresses that the proposition is true for (k+1). Hence, if the proposition is true for any particular number, it must be true for the next higher number.

Evidently the proposition is true for k = 1, hence it is true for k = 2, and therefore again it is true for k=3, and so on for higher and higher numbers.

As this mode of increasing k can never reach an end, the proposition is generally true.