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304 ALGEBRA. Find the number of terms and the common difference when

31. The first term is 3, the last term 90, and the sum 1395. 32. The first term is 79, the last term 7, and the sum 1075. 33. The sum is 24, the first term 9, the last term —6. 34. The sum is 714, the first term 1, the last term 58 1 2. 35. The last term is —16, the sum —133, the first term —3. 36. The first term is —75, the sum —740, the last term 1. 37. The first term is a, the last 13a, and the sum 49a. 38. The sum is —320x, the first term 3x, the last term —35x.

369. If any two terms of an Arithmetical Progression be given, the series can be completely determined; for the data furnish two simultaneous equations, the solution of which will give the first term and the common difference. Ex. Find the series whose 7th and 51st terms are —3 and —355 respectively.

If a be the first term, and d the common difference, —3 = the 7th term = « + 6 fZ ; and —355 = the 51st term = a + 50 cZ ; whence, by subtraction, —352 = 44 fZ ; . • . fZ = —8 ; and, consequently, a = 45. Hence the series is 45, 37, 29 •••.

370. Arithmetic Mean. When three quantities are in Arithmetical Progression, the middle one is said to be the arithmetic mean of the other two.

Thus a is the arithmetic mean between a — d and a + d.

371. To find the arithmetic mean between two given quantities.

Let a and b be the two quantities; A the arithmetic mean. Then, since a, A, b, are in A.P., we must have

b — A = A — a,

each being equal to the common difference;

whence A = a + b 2