EXAMPLES. 1. The first term of an increasing arithmetical series is 3, the common difference 2, and the number of terms 20; required the sum of the series. First, 3+2(20-1)=3+2×19=3+38=41, the last 20 Or, {2×3+(20−1)×2}x=(6+19×2)×10=(6+ 38)×10=44×10=440, as before. 2. The first term of a decreasing arithmetical series is 100, the common difference 3, and the number of terms 34; required the sum of the series. First, 100-3(34-1)=100-3×33=100-99=1, the last term. 34 Or, 2×100-(34-1) × 3}× = (200-33 × 3) ×17=(200-99)×17=101×17=1717, as before. 3. Required the sum of the natural numbers, 1, 2, 3, 4, 5, 6, &c. continued to 1000 terms. Ans. 500500. 4. Required the sum of the odd numbers 1, 3, 5, 7, 9, &c. continued to 101 terms. Ans 10201. 5. How many strokes do the clocks of Venice, which go on to 24 o'clock, strike in a day? Ans. 300. Where the upper sign + is to be used when the series is 'ncreasing, and the lower sign - when it is decreasing; also the last term la±(n-1)d, as above. 6. Required the 365th term of the series of even numbers 2, 4, 6, 8, 10, 12, &c. Ans. 720. 7. The first term of a decreasing arithmetical series is 1 10, the common difference and the number of terms 3' 21; required the sum of the series. Ans. 140. 8. One hundred stones being placed on the ground, in a straight line, at the distance of a yard from each other; how far will a person travel, who shall bring them one by one, to a basket, placed at the distance of a yard from the first stone ? Ans. 5 miles and 1300 yards. GEOMETRICAL PROPORTION, is the relation which two quantities of the same kind have to two others, when the antecedents, or leading terms of each pair, are the same parts of their consequents, or the consequent of the antecedents. And if two quantities only are to be compared together, the part, or parts, which the antecedent is of its consequent, or the consequent of the antecedent, is called the ratio; observing, in both cases, always to follow the same method. Hence, three quantities are said to be in geometrical proportion, when the first is the same part, or multiple, of the second, as the second is of the third. Thus, 3, 6, 12, and a, ar, ara, are quantities in geometrical proportion. And four quantities are said to be in geometrical proportion, when the first is the same part, or multiple, of the second, as the third is of the fourth. Thus, 2, 8, 3, 12, and a, ar, b, br, are geometrical proportionals. Direct proportion, is when the same relation subsists between the first of four terms and the second, as between the third and fourth. Thus, 3, 6, 5, 10, and a, ar, b, br, are in direct proportion. Inverse, or reciprocal proportion, is when the first and second of four quantities are directly proportional to the reciprocals of the third and fourth Thus, 2, 6, 9, 3, and a, ar, br, b, are inversely pro 11 and a, ar, brb are direct GEOMETRICAL PROGRESSION is when a series of quantities have the same constant ratio; or which increase, or decrease, by a common multiplier, or divisor. Thus, 2, 4, 8, 16, 32, 64, &c. and a, ar, ar2, ar3, ar4, &c. are series in geometrical progression. The most useful properties of geometrical proportion and progression are contained in the following theo rems : 1. If three quantities be in geometrical proportion, the product of the two extremes will be equal to the square of the mean. Thus if the proportionals be 2, 4, 8, or a, b, c, then will 2×8=42, and aXc=ba. 2. Hence, a geomètrical mean proportional, between any two quantities, is equal to the square root of their product. Thus, a geometric mean between 4 and 9 is = 36 6. And a geometric mean between a and b is = ab. 3. If four quantities be in geometrical proportion, the product of the two extremes will be equal to that of the means. H2 Thus, if the proportionals be 2, 4, 6, 12, or a, b, e, d; then will 2×12=4×6, and a Xd=bXc. 4. Hence, the product of the means of four proportional quantities, divided by either of the extremes, will give the other extreme; and the product of the extremes, divided by either of the means, will give the other mean. 9×5 3×15 bXc = d, Thus, if the proportionals be 3, 9, 5, 15, or a, b, c, d; then will =15, and =9: also, axd and G =b. a 5. Also, if any two products be equal to each other, either of the terms of one of them, will be to either of the terms of the other, as the remaining term of the last is to the remaining term of the first. Thus, if ad=bc, or 2×15=6×5, then will any of the following forms of these quantities be proportional: › Directly, a : b :: c:d, or 2:6:: 5:15. Invertedly, b:a::d:c, or 6 : 2 :: 15:5. Alternately, a:c::b:d, or 2:5::6:15. Conjunctly, a: a+b :: c:c+d, or 2:8::5:20. Disjunctly, a: bra :: c: dsc, or 2:4::5:10. Mixedly, b+a: bra :: d+c: dc, or 8:4::20:10. In all of which cases, the product of the two extremes is equal to that of the two means. 6 In any continued geometrical series, the product of the two extremes is equal to the product of any two means that are equally distant from them; or to the square of the mean, when the number of terms is odd. Thus, if the series be 2, 4, 8, 16, 32; then will 2×32=4×16=82 7. In any geometrical series, the last term is equal to the product arising from multiplying the first term by such a power of the ratio as is denoted by the number of terms less one. Thus, in the series 2, 6, 18, 54, 162, we shall have 2×34=2×81=162. And in the series a, ar, ar2, ar3, ar4, &c. continued to n terms, the last term will be l=arn-1. 8. The sum of any series of quantities in geometrical progression, either increasing or decreasing, is found by multiplying the last term by the ratio, and then dividing the difference of this product and the first term by the difference between the ratio and unity. • Thus, in the series 2, 4, 8, 16, 32, 64, 128, 256, 512, 512×2-2 we shall have = 1024-2=1022, the sum of 2-1 the terms. Or the same rule, without considering the last term, may be expressed thus : Find such a power of the ratio as is denoted by the number of terms of the series; then divide the difference between this power and unity, by the difference between the ratio and unity, and the result, multiplied by the first term, will be the sum of the series. Thus, in the series a+ar+ar2 + ar3 + ar2, &c. to arn-1, we shall have S=a() Where it is to be observed, that if the ratio, or common multiplier, r, in this last series, be a proper fraction, and consequently the series a decreasing one, we shall have, in that case, a+ar+ar2 + ar3+ar, &c. ad infinitum = a 1 |