2. Expand1+1=2, into an infinite series. Ans 1+-+10-155 &c. 3. Expand 1-1 into an infinite series. Ans. 1----TT&C. 4. Expand✔ a2 + x into an infinite series. PROBLEM III. To Extract any Root of a Binomial: or to Reduce a Binomial Surd into an Infinite Series. THIS will be done by substituting the particular letters of the binomial, with their proper signs, in the following general theorem or formula, viz. m (P + PQ) m-2n 2n the and it will give the root required: observing that P denotes the first term, the second term divided by the first, index of the power or root; and ▲, B, C, D, &c., denote the several foregoing terms with their proper signs. EXAMPLES. 1. To extract the sq. root of a2 + b2, in an infinite series. = (a2) = (a2) AQ = · Χα Χ b2 62 and b2 —=— a3 2a a = A, the 1st term of the series. =B, the 2d term. m-n 1-2 b2 ba 64 BQ X = c, the 3d term. Note. To facilitate the application of the rule to fractional exampies, it is proper to observe, that any surd may be taken from the denominator of a fraction and placed in the numerator, and vice versa, by only changing the sign of its index. Thus, 3. Find the value of ✨/ (a3 — b3) or (a3 —b3)3 in a series. Ans. a 63 bo 56° Заа 9a5 81a &c. 8. To find the value of §/ (a3+x3) or (a3 +x3) in a series. ARITHMETICAL PROPORTION is the relation between two numbers with respect to their difference. Four quantities are in Arithmetical Proportion, when the difference between the first and second is equal to the difference, between the third and fourth. Thus, 4, 6, 7, 9, and a, a +d, b, b x d are in arithmetical proportion. Arithmetical Progression is when a series of quantities have all the same common difference, or when they either increase or decrease by the same common difference. Thus 2, 4, 6, 8, 10, 12, &c. are in arithmetical progression having the common difference 2; and a, a +d, a + 2d, a + 3d, a + 4d, a + 5d, &c. are series in arithmetical progression, the common difference being d. The most useful part of arithmetical proportion is contained in the following theorems : 1. When four quantities are in Arithmetical Proportion, the sum of the two extremes is equal to the sum of the two means. Thus, in the arithmetical 4, 6, 7, 9, the sum 4 + 9 = 6 + 7 = 13: and in the arithmeticals a, a + d, b, b + d, the sum a+b+ d = a + b + d. 2. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two terms at an equal distance from them. Thus, Thus, if the series be 1, 3, 5, 7, 9, 11, &c. 3. The last term of any increasing arithmetical series, is equal to the first term increased by the product of the common difference multiplied by the number of terms less one; but in a decreasing series, the last term is equal to the first term lessened by the said product. Thus, the 20th term of the series, 1, 3, 5, 7, 9, &c. is = 1 + 2 (20−1) = 1 + 2 × 19 = 1 +38 = 39. And the nth term of a, a-d, a- 2d, a-3d, a - 4d, &c. is a-(n-1) xda-(n-1) d. 4. The sum of all the terms in any series in arithmetical progression, is equal to half the sum of the two extremes multiplied by the number of terms. Thus, the sum of 1, 3, 5, 7, 9, &c. continued to the 10th (1 +19) X 10 20 X 10 term, is = 2 = 2 = 10 X 10 = 100. And the sum of n terms of a, a + d, a + 2d, a + 3d, to 1. The first term of an increasing arithmetical series is 1, the common difference 2, and the number of terms 21; required the sum of the series ? First, 12 x 20 1 +40=41, is the last term, × 20 = 21 X 20=420, the sum required, 2. The first term of a decreasing arithmetical series is 199, the common difference 3, and the number of terms 67; re quired the sum of the series? 3. To find the sum of 100 terms of the natural numbers 1, 2, 3, 4, 5, 6, &c. Ans. 5050. 4. Required |