S=nt 2+ n(n-1) n(n-1)(n-2) 2 2 · 3 all the terms after the third becoming zero. By performing the indicated operations, adding the results, and then factoring, we have n(n+1)(n+2), Ans. S= 6 EXAMPLES FOR PRACTICE. 1. Find the 9th term of the series 1, 4, 8, 13, 19, eto. Ans. 53. 2. Find the 15th term of the series 1, 4, 10, 20, 35, etc. Ans. 680. 3. Find the 8th and 9th terms of the series 1, 6, 21, 56, 126, 251, 456, etc. Ans. 781 and 1231. 4. Find the 20th term of the series 1, 8, 27, 64, 125, etc. Ans. 8000. 5. Find the nth term of the series 1, 3, 6, 10, 15, 21, etc. n(n+1) Ans. 2 6. Find the nth term of the series 1, 4, 10, 20, 35, etc. Ans. 6 Ans. 24 8. Sum the series 1, 3, 6, 10, 15, 21, etc., to 20 terms. Ans. 1540. 9. Sum the series 1, 5, 14, 30, 55, 91, etc., to 12 terms. Ans. 2366. 10. Sum the series 1, 4, 13, 37, 85, 166, etc., to 10 terms. Ans. 2755. 11. Sum the series, 1:2, 23, 3 · 4, 4.5, 5.6, etc., to n terms n(n+1)(n+2), Ans. 3 12. Sum the series 1.2.3, 2.3.4, 3.4.5, 4.5.6, etc., to n terins. n(n+1)(n+2)(n+3) Ans. 4 13. Sum the series 1o, 2o, 3o, 4o, 5', etc., to n terms. n(n+1)(2n+1) Ans. 6 14. Sum the series 1o, 2o, 3o, 4', 5', etc., to n terms. Ans. Ans. (****) 15. Sum the series 1, 2, 3, 4, 5*, eto., to n terms. no 5 +ģ n 30 16. Sum the series (m+1), 2(m+2), 3(m+3), 4(m+4), etc., to n terms. n(n+1)(1+2n+3m) Ans. 6 INTERPOLATION. 396. Interpolation is the process of introducing between the terms of a series, intermediate terms which shall conform to the law of the series. It is of great use in the construction of mathematical tables, and in the calculations of Astronomy. 397. The interpolation of terms in a series is effected by the differential method. In any series, the value of a term which has n terms before it is expressed by formula (m), (393), which is n(n-1) n(n-1)(n-2) Tith = a + nd, t dat dz+.... 2 2 3 If in this formula we make n a fraction, then the resulting equation will give the value of a term intermediate between two of the given terms, and related to the others by the law of the series. If n is less than unity, the intermediate term will lie between the first and second of the given terms; if n is greater than 1 and less than 2, the intermediate term will lie between the second and third of the given terms; and so on. 821 = 2.758924 0/22 = 2.802039 to find the cube roots Given P23 = 2.843867 > of intermediate num024 = 2.884501 bers, by interpolation. 325 2.924018 1. Required the cube root of 21.75. We have 21 2.758924 22 2.802039 +.043115 23 2.813867 +.041828 -.001287 24 | 2.884501 +:040632 --.001196 +.000091 25 2.924018 +.039519 –.001113 +.000083 -.000008 Hence, to find the cube root of 21.75 by the formula, we have a = 2.758924, n = .75, di = +.043115, d2 = –.001287, dz = +.000091, eto. These values substituted in the formula, give 1st term, +2.758924 + .032336 3d + .000121 4th + .000004 Whence, 2.791385, Ans. If it were required to find the cube root of any number between 22 and 23, we might put n equal to the excess of the number above 21, and employ the same values for dy, dz, dz, etc., as before. But greater accuracy will be attained by making 22 the first term of the series, and employing the corresponding differences ; in which case n will be a proper fraction. 2d EXAMPLES FOR PRACTICE. Find by interpolation, Ans. 2.773083. 3. The cube root of 21.4568. Ans. 2.778785. 4. The cube root of 22.25. Ans. 2.812613. 5. The cube root of 22.684, Ans. 2.830784. 6. The cube root of 22.75. Ans. 2.833525. 398. On three consecutive days, the angular distances of the sun from the moon, as seen from the earth, were as follows: 1st day, noon, 66° 6' 38". 78° 34' 48". 84° 39' 4". 90° 37' 18". 96° 29' 57". In the data here given, the interval of time is 12 hours. Hence, to find the distance of the sun from the moon at intermediate times, n must always be some fractional part of 12. Thus, for the distance at 3 o'clock P. M. of the first day we have n= = , and u = 66° 6' 38''; for the distance at 6 o'clock A. M. of the second day, = j, and a = 72° 24' 5''. For the distance at 3 o'clock P. M. of the second day, n = = , and a = 78° 34' 48". EXAMPLES FOR PRACTICE. Find by interpolation the distance of the sun from the moon, 1. At 3 o'clock P. M. of the first day. Ans. 67° 41' 38". 2. At 6 o'clock P. M. of the first day. Ans. 69° 16' 13". 3. At 9 o'clock P. M. of the first day. Ans. 70° 50' 21". 4. At 3 o'clock A. M. of the second day. Ans. 73° 57' 23". 5. At 6 o'clock A. M. of the second day. Ans. 75° 30' 16". 6. At 9 o'clock A. M. of the second day. Ans. 77° 2' 44". 7. At 3 o'clock P. M. of the second day. Ans. 80° 6' 27". 8. At 6 o'clock P. M. of the second day. Ans. 81° 37' 43". 9. At 9 o'clock P. M. of the second day. Ans. 83° 8' 35" LOGARITHMS. 399. The Logarithm of a number is the exponent of the power to which a certain other number, called the base, must be raised, in order to produce the given number. Thus, in the expression, a = b, the exponent, x, is the logarithm of 6 to the base a. An equation in this form is called an exponential equation. If in this equation we suppose a to be constant, while 6 is made equal to every possible number in succession, the corresponding values of x will constitute a system of logarithms: hence, 400. A System of Logarithms consists of the logarithms of all possible numbers, according to a given base. Any positive number greater than unity may be made the base of a system of logarithms. For, by giving to x suitable values, the equation a = b will be true for all possible values of b, provided a is positive and greater than 1. Hence, There may be an indefinite number of systems of logarithms. 401. If in the equation a* = 1, we suppose b to represent a perfect power of a, then x will be some integer ; but if b is not a perfect power of a, then x will be some fraction. Hence, A logarithm may consist of an integral and a fractional part. 402. The Index or Characteristic of a logarithm is the integral part; and 403. The Mantissa is the fractional part of a logarithm. For illustration, let 5 be the base of a system; then we have 52.25 = 51 = V5° = 37.384. Thus, the logarithm of 37.384 to the base 5, is 2.25; the index of this logarithm is 2, and the mantissa .25. PROPERTIES OF LOGARITHMS. 404. There are certain properties of logarithms, which are common to all systems. To investigate these general properties, let |