EXAMPLES IN ARITHMETICAL AND GEOMETRICAL PROGRESSIONS. 1. Find the sum of 21 terms of the series 1, 3, 5, 7, &c. Ans. 441. 2. Find the sum of 37 terms of the series 199, 196, 193, 190, &c. Ans. 5365. 3. The sum of an arithmetical progression is 45, common difference, and the number of terms 13: find the first term. Ans. 2. 4. The sum of an arithmetical progression is 88, the first term 3, and the number of terms 16: find the common difference. Ans.. 5. The sum of an arithmetical progression is 567, the first term 7, and common difference 2: find the number of terms. Ans. 21. 6. Find the sum of 30 terms of the series 8, 11, 14, 17, &c. Ans. 1545. 7. Find the 23rd term of the progression 29, 32, 35, &c. Ans. 95. 8. Find 2 geometrical means between 3 and 192. Ans. 12, 48. 9. Find the exact value of 1666 of a pound. Ans. 3s. 4d. 14. Ans. 74, 74, 7, 7, 7, 8, 81. Find 4 numbers in arithmetical progression, that their sum may be 16, and the sum of their squares 84. Ans. 1, 3, 5, 7. 15. One hundred stones being placed on the ground in a straight line, at a distance of 2 yards from each other; how far must a person go to put them one by one into a basket, which is placed 2 yards from the first stone? Ans. 11 miles 840 yards. 16. Two men, A and B, set out at the same time in the same direction: A travels 8 miles a day; B travels the first day 1 mile, the second 2 miles, the third 3 miles, &c. In how many days will B overtake A ? 17. Ans. 15. Find 4 numbers in arithmetical progression, that the product of the means may be 12, and the sum of the extremes 7. Ans. 2, 3, 4, 5. 18. A man agreed to work for a year at the rate of a farthing for the 1st day, a half-penny the 2nd, three farthings the 3rd, and so on. What money did he receive at the year's end? Ans. 697. 11s. 6d. 19. The sum of 3 numbers in geometrical progression is 65, and the first is 5: find the two last. Ans. 15, 45. 20. The sum of the first and second of 3 numbers in geometrical progression is 24, and the difference of the second and third is 36: find the numbers. Ans. 6, 18, 54. FRACTIONAL INDICES AND SURDS. 106. Although mention has been made of fractional indices or exponents (Art. 10, 50) yet the operations have been hitherto confined, with the exception of the square root, to integer ones. From the mode of expressing the powers of quantities, it has been concluded immediately from the definition (Art. 9) that the multiplication or division of the quantities may be effected by the addition or subtraction of their indices; that is, if m and n are integer and positive, and m greater than n, then algebraically speaking Therefore, any quantity raised to the power of O is equal to unity. 107. The powers and roots of quantities may be expressed by negative as well as by positive indices. A quantity therefore may be removed from the numerator of a fraction into the denominator, or from the denominator into the numerator, by changing the sign of its index; and a" is only 108. Also aman-am+n when m and n are in 109. The theorem amnam+" is equally true when the indices are fractional; observing that the denominator of the index denotes the root to |