~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? 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~