Part 28

3. In the G. P. 2, 6, 18, ..., which term is 486?

4. Find x, if 2x - 4, 5x - 7, 10x + 4 are in geometrical progression.

5. How can you turn a G. P. into an equation?

6. Insert 4 geometrical means between 4 and 972.

7. Insert 6 geometrical means between 5/16 and 5120.

8. Given a = -2, n = 5, l = -32; find r and S.

9. If the first term of a geometrical progression is 12 and the sum to infinity is 36, find the 4th term.

10. If the series 3-1/3, 2-1/2, ... be an A. P., find the 97th term. If a G. P., find the sum to infinity.

11. The third term of a geometrical progression is 36; the 6th term is 972. Find the first and second terms.

12. Insert between 6 and 16 two numbers, such that the first three of the four shall be in arithmetical progression, and the last three in geometrical progression.

13. A rubber ball falls from a height of 40 inches and on each rebound rises 40% of the previous height. Find by formula how far it falls on its eighth descent. (Yale.)

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

~THE BINOMIAL THEOREM~

1. Review the Binomial Theorem laws. (See Involution.)

Expand:

2. (b - n)^7.

3. (x + x^(-1))^5.

4. [a/x - x/a]^6.

5. [x/2y - ^(1/2)]^5.

6. (x^2 - x + 2)^3.

7. [(2[b^2]^(1/3))/(y) + (3[y^(1/2)])/(b^3)]^4.

8. (a + b)^n = a^n + na^(n - 1)b + [n(n - 1)]/(1.2) a^(n - 2)b^2 + [n(n - 1)(n - 2)]/(1.2.3) a^(n - 3)b^3 + [n(n - 1)(n - 2)(n - 3)]/(1.2.3.4) a^(n - 4) b^4 + ....

Show by observation that the formula for the

(r + 1)th term = [n(n - 1)(n - 2)...(n - r + 1)]/[1.2.3.4 ... r] a^(n - r)b^r.

9. Indicate what the 97th term of (a + b)^n would be.

10. Using the expansion of (a + b)^n in (8), derive a formula for the rth term by observing how each term is made up, then generalizing.

Using either the formula in (8) or (10), whichever you are familiar with, find:

11. The 4th term of [a + 1/a]^(30).