where the coefficients of the value of b, the second term in the given series, are the coefficients of the first power of a binomial; the coefficients of the value of c, the third term, are the coefficients of the second power of a binomial; and so on. Hence, we infer that the coefficients of the nth term of the series are the coefficients of the (n − 1)th power of a binomial; whence, representing the nth term of the series by T, we shall have

When the differences at last become 0, the value of the nth term can be obtained exactly; but, in other cases, the result will be merely an approximate value.

385. To find the sum of any number of terms of a series. Let the series be

α, b, c, d, e,

Assume the series

(1)

(2)

0, 0+a, a + b, a+b+c, a+b+c+d,..... where the (n+1)th term is obviously equal to the sum of n terms in the proposed series, and the first order of differences is equal to series (1). Hence, the (n + 1)th order of differences in series (2) is the nth order in series (1). If, then, we substitute in formula (A), 0 for a, n+1 for n, a for d1, d1 for d2, etc., and denote the sum of n terms of the proposed series by S, we shall obtain

Then, by substituting in formula (A), we have for the 12th term

2+(12—1)4+

(12—1) (12—2) 2—2+44+110=156, Ans.

1.2

2. Find the sum of 8 terms of the series 2, 5, 10, 17,..... Here, we find a = 2, di 3, d2: =2, and n 8.

Then, by substituting in formula (B), we have

3. Find the first term of the fifth order of differences of the series 6, 9, 17, 35, 63, 99, . . .

Ans. 3.

4. Find the first term of the sixth order of differences of the series 3, 6, 11, 17, 24, 36, 50, 72,

5. Find the seventh term of the series 3, 5, 8, 12, 17,

8. Find the 15th term of the series 12, 22, 32, 42,

...

Ans. 225.

9. Sum 18+23 + 33 + 43 + 53 +.......... to the nth term. Ans. (n+2 n3 + n2).

11. If shot be piled in the shape of a pyramid, with a triangular base, each side of which exhibits 9 shot, find the number contained in the pile. Ans. 165.

12. If shot be piled in the shape of a pyramid, with a square base, each side of which exhibits 25 shot, find the number contained in the pile. Ans. 5525.

INTERPOLATION.

386. INTERPOLATION is the process of introducing between terms of a series other terms conforming to the law of the series.

Its usual application is in finding intermediate numbers between those given in Mathematical Tables, which may be regarded as a series of equidistant terms.

387. The interpolation of any intermediate term in a series, is essentially finding the nth term of the series, by the differential method (Art. 384). Thus,

Let t represent the term to be interpolated in a series of equidistant terms, and p the distance the term t is removed from the first term, a, that is, p = n — 1.

Then, by substituting p for n-1 in formula (A), Art. 384, we have the formula of interpolation,

p −

t=a+pd2+P(p—1) d2+P (p − 1) (p −2)

1.2

1.2.3

ds+..

EXAMPLES.

1. In the series,,, 16, 17,

find the

middle term between and

Since each interval between the terms is to be reckoned as unity, the distance from the first term to the required middle term is 21⁄2 intervals, or p= 23.

.

Make a= 13, the denominator of the first term; then, by the preceding formula,

13+2 × 1 = 15, and 1÷15 = 31,

or the proposed middle term.

=

2. Given the square root of 94 9.69536, of 95

9.74679, and of 96 =

of 941.

9.79796, to find the square root

Since the distance is of the first interval, p=.

Hence, 941 =

9.69536(05143)-(-.00026)

= 9.69536.01286+.00002=9.70824, Ans.

3. Given the cube root of 64 = 4, of 65 - 4.0207, of 664.0412, and of 67 = 4.0615, to find the cube root of 66.5. Ans. 4.0514.

4. Required the number of miles in a degree of longitude in latitude 42° 30', the length of a degree of longitude

in latitude 41° being 45.28 miles, in latitude 42° being 44.59 miles, in latitude 43°, 43.88 miles, and in latitude 44°, 43.16 miles. Ans. 44.24 miles.

3.556893, of 47 =

5. Given the cube root of 45 3.608826, of 49 = 3.659306, and of 51= 3.708430, to find the cube root of 48.

Ans. 3.634241.

6. If the amount of $1 at 7 per cent compound interest for 2 years is $1.145, for 3 years is $1.225, for 4 years $1.311, and for 5 years $1.403, what is the amount for 4 years and 6 months? Ans. $1.357.

LOGARITHMS.

388. The LOGARITHM of a number is the exponent of the power to which some fixed number, called the base, must be raised, to produce the given number. Thus, suppose

a2 = n,

then x is the logarithm of n to the base a, and may be written

where log is read logarithm to base a. The subscript is usually omitted when the base is readily understood.

389. If, in the preceding equation, a remaining fixed, n be supposed to assume in succession all positive values, the corresponding values of x, taken together, will constitute a system of logarithms.

But, since a may be made any positive number greater than unity, there may be different systems of logarithms.