Part 27

4. 2a + 3c : 2a - 3c = 8b + 12d : 8b - 12d.

5. a^2 - ab + b^2 : (a^3 - b^3)/a = c^2 - cd + d^2 : (c^3 - d^3)/c.

6. The second of three numbers is a mean proportional between the other two. The third number exceeds the sum of the other two by 20; and the sum of the first and third exceeds three times the second by 4. Find the numbers.

7. Three numbers are proportional to 5, 7, and 9; and their sum is 14. Find the numbers. (College Entrance Board.)

8. A triangular field has the sides 15, 18, and 27 rods, respectively. Find the dimensions of a similar field having 4 times the area.

~ARITHMETICAL PROGRESSION~

1. Define an arithmetical progression.

Learn to derive the three formulas in arithmetical progression:

l = a + (n - 1)d, S = (n/2)(a + l), S = (n/2)[2a + (n - 1)d].

2. Find the sum of the first 50 odd numbers.

3. In the series 2, 5, 8, ..., which term is 92?

4. How many terms must be taken from the series 3, 5, 7, ..., to make a total of 255?

5. Insert 5 arithmetical means between 11 and 32.

6. Insert 9 arithmetical means between 7-1/2 and 30.

7. Find x, if 3 + 2x, 5 + 6x, 9 + 5x are in A. P.

8. The 7th term of an arithmetical progression is 17, and the 13th term is 59. Find the 4th term.

9. How can you turn an A. P. into an equation?

10. Given a = -5/3, n = 20, S = -5/3, find d and l.

11. Find the sum of the first n odd numbers.

12. An arithmetical progression consists of 21 terms. The sum of the three terms in the middle is 129; the sum of the last three terms is 237. Find the series. (Look up the short method for such problems.) (Mass. Inst. of Technology.)

13. B travels 3 miles the first day, 7 miles the second day, 11 miles the third day, etc. In how many days will B overtake A who started from the same point 8 days in advance and who travels uniformly 15 miles a day?

~Reference:~ The chapter on Arithmetical Progression in any algebra.

~GEOMETRICAL PROGRESSION~

1. Define a geometrical progression.

Learn to derive the four formulas in geometrical progression:

{ I. l = ar^(n - 1). {II. S = (ar^n - a)/(r - 1).

{III. S = (rl - a)/(r - 1). { IV. S_{[infinity]} = (a)/(1 - r).

2. How many terms must be taken from the series 9, 18, 36, ... to make a total of 567?