5. Find the value of x++ in an infinite series. Ans. x's+ α α b+c 6. To find in an infinite series. Ans. — × 7. Find a2+b in an infinite series. Ans. a+; + 2a 8 a3 8. Extract the 5th root of 248832 by infinite series. Ans. 12. 14. A series being given, to find the several orders of differences. RULE I. Subtract the first term from the second, the second from the third, the third from the fourth, and so on; the several remainders will constitute a new series, called the first order of differences. II. In this new series, take the first term from the second, the second from the third, &c. as before, and the remainders will form another new series, called the second order of differ ences. III. Proceed in the same manner for the third, fourth, fifth, &c. orders, until either the differences become 0, or the work be carried as far as is thought necessary . d Let a, b, c, d, e, &c. be the terms of a given series, then if D=the first term of the nth order of differences, the following theorem will exhibit the value n-1n-2 of D: viz. ±a+nb±n." .c+n!" ・.d+n.· 23 n-in-2 n-3 •.eF, &c. (to n + 1 terms)=D, where the upper signs must be taken when n is an even number, and the lower signs when n is odd. EXAMPLES.-1. Given the series 1, 4, 8, 13, 19, 26, &c. to find the several orders of differences. Thus 1, 4, 8, 13, 19, 26, &c. the given series. Then... And. ... ....... 0, 0, 0, &c. the third differences. where the work evidently must terminate. 2. Given the series 1, 4, 8, 16, 32, 64, 128, &c. to find the several orders of differences. Here 1, 4, 8, 16, 32, 64, 128, &c. given series. 3. Find the several orders of differences in the series 1, 2, 3, 4, &c. Ans. First differences 1, 1, 1, 1, &c. Second diff. 0, 0, 0, &c. 4. To find the several orders of differences in the series 1, 4, 9, 16, 25, &c. Ans. First differences 3, 5, 7, 9, &c. Second 2, 2, 2, &c. Third 0, 0, &c. 5. Required the orders of differences in the series 1, 8, 27, 64, 125, &c. 6. Given 1, 6, 20, 50, 105, &c. to find the several orders of differences. 7. Given the series 1, 3, 7, 13, 21, &c. to find the third and fourth orders of differences. 15. To find any term of a given series. RULE I. Let a, b, c, d, e, &c. be the given series; d', d11, d111, dir, &c. respectively, the first term of the first, second, third, fourth, &c. order of differences, as found by the preceding article; n=the number denoting the place of the term required. If the differences be very great, the logarithms of the quantities may be used, the differences of which will be much smaller than those of the quantities themselves; and at the close of the operation the natural number answering to the logarithmical result will be the answer. See Emerson's Differential Method, prop. 1. EXAMPLES.-1. To find the 10th term of the series 2, 5, 9, Where d1=3, d11=1, d=0, also a=2, n=10; wherefore 1=)2+27+36=65=the 10th term required. 2. To find the 20th term of the series 2, 6, 12, 20, 30, &c. Here a=2, n=20; and Art. 12. 2, 2, 2, &c, 2nd diff. or d1=4, d11=2; whence +342=420=the 20th term required. 3. Required the 5th term of the series 1, 3, 6, 10, &e. Ans. 15. 4. To find the 10th term of the series 1, 4, 8, 13, 19, &c. Ans. 64. 5. To find the 14th term of the series 3, 7, 12, 18, 25, &c. Ans. 133. 6. Required the 20th term of the series 1, 8, 27, 64, 125, &c. Ans. 8000. 7. To find the 50th term of 1, 4, 8, 13, 19, &c. 8. To find the 10th term of 3, 7, 12, 18, 25, &c. 16. If the succeeding terms of a given series be at an unit's distance from each other, any intermediate term may be found by interpolation, as follows. • For the investigation of this rule, see Emerson's Differential Method, prop. 2. RULE I. Let y be the term to be interpolated, x its distance from the beginning of the series, d1, d11, d111, div, &c. the first terms of the several orders of differences. EXAMPLES.-1. Given the logarithms of 105, 106, 107, 108, and 109, to find the logarithm of 107.5. 5 2 2 Here x=(107.5-105=2.5) =the distance of the term y, a=.0211893, d1=41166, d11=—387, d1111=—8, div——2. Then y=a+xd1 +x.- -.d11+x: x-2x-3 · d3 = (a + 2/2 3 4 5 3 1 2 X X × dill+ 2 4 6 1x- -2 2 3 5 3 1 2 16 128 0211893+102915-725-2.5+.078=.031407128, the logarithm 2. Given the logarithmic sines of 3° 41, 3° 51, 3° 61, 3o 71, and 3o 8', to find the sine of 3o 61 1511. Herex (3° 6' 1511-3°41=2° 151=) termy, to be interpolated; a=8,7283366, d'=23516, d11——126, ↑ This rule is investigated in Emerson's Differential Method, prop. 5. VOL. II. 4. Given the logarithmic sines of 1o 01, 1o 1', 1o 21, and 1o 3', to find the logarithmic sine of 1o 11 4011. Ans. 8.2537533. 17. If the first differences of a series of equidifferent terms be small, any intermediate term may be found by interpolation, as follows. RULE I. Let a, b, c, d, e, &c. represent the given series, and n=the number of terms given. n-3 4 .e+, &c.zo, from whence, by transposition, &c. any re quired term may be obtained &. EXAMPLES.-1. Given the square root of 10, 11, 12, 13, and 15, to find the square root of 14. Here n=5, and e is the term required. a=(✓✓/10=)3.1622776 b=(11)3.3166248 c=(12)3.4641016 d=(13)3.6055512 f=(15)3.8729833 n-1 And since n=5, the series must be continued to 6 terms. Therefore anb+n.- ..c-n. n-in-2 n-1n-2 -.d+n. 2 3 For the investigation of this rule, see Emerson's Differential Method, prop. 6. |