OF ARITHMETICAL PROPORTION AND PROGRESSION. ARITHMETICAL PROPORTION, is the relation which two quantities, of the same kind, have to two others, when the difference of the first pair is equal to that of the second. Hence, three quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the second and third. Thus, 2, 4, 6, and a, a+b, a +-26, are quantities in arithmetical proportion. And four quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth. Thus, 3, 7, 12, 16, and a, atb, c, c+b, are quantities in aritbmetical proportion. ARITHMETICAL PROGRESSION is when a series of quantities increase or decrease by the same common diffe Thus, 1, 3, 5, 7, 9, &c. and a, atd, a+2d, a +3d, &c. are increasing series in arithmetical progression, the common differences of which are 2 and d. Aud 15, 12, 9, 6, 8.c. and a, a-d, a-2d, a-3d, &c. are decreasing series in arithmetical progression, the common differences of which are 3 and d. The most useful properties of arithmetical proportion and progression are contained in the following theorems : 1. If four quantities are in arithmetical proportion, the sum of the two extremes will be equal to the sum of the two means. Thus, if, the proportionals be 2, 5, 7, 10, or a, b, c, d ; then will 2+10=5+7, and a+d=6+c. 2. And if three quantities be in arithmetical propor rence. tion, the sum of the two extremes will be double the mean. Thus, if the proportionals be 3, 6, 9, or a, b, c, then will 3+9=2x6 =12, and ate=2b. 3. Hence an arithmetical mean between any two quantities is equal to half the sum of those quantities. Thus, an arithmetical mean between 2 and 4 is 2+4 5+6 =3; and between 5 and 6 it is = =51. 2 2 at6 And an arithmetical mean between a and 6 is 2 4. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two terms that are equally distant from them, or to double the middle term, when the number of terms is odd. Thus, if the series be 2, 4, 6, 8, 10, then will 2+10 =4+8=2X6=12. And, if the series be a, atd, a+2d, a +3d, a +4d, then will a+(at4d)=(a+d)+(a+30)=2x(a+2d). 5. The last term of any increasing arithmetical series is equal to the first term plus the product of the common difference by the number of terms less one ; and if the series be decreasing, it will be equal to the first term minus that product. Thus, the nth term of the series a, atd, at2d, at3d, a+4d, &c. is a+(n-1)d. And the nth term of the series a, a-d, a-2d, a- - 3d, a-4d, &c. is a-(1-1)d. 6. The sum of any series of quantities in arithmetical progression is equal to the sum of the two extremes multiplied by half the number of terms. Thus, the sum of 2, 4, 6, 8, 10, 12, is = (2+12) 6 =14X3=42. And if the series be at (4+d)+(at2d)+(2+3d) we +(a+4d) &c. tl. and its sua; he denoted by S, shall have S=(a +1)*, where l is the last term, and ni the number of terms. Or, the sum of any increasing arithmetical series may be found, without considering the last term, hy adding the product of the common difference by the number of terms less one to twice the first term, and then multiply. ing the result by half the number of teruis. And, if the series be decreasing, its sum will be found by subtracting the above product from twice the first term, and then multiplying the result by half the number of terms. as hefore. Thus, if the series be a+u+d+a+20)+(a+3d) + (a + 4d), &c. continued to n terms, we shall have S= X And if the series be a+(a-d)+(a-2d)+(a-3d) + (a-4d), &c. to n terms, we shall have s={2a—(n-1)d)x= (y). n 2a+(n-1)a} n (y) The sum of any number of terms (n) of the series of natural numbers 1, 2, 3, 4, 5, 6, 7, &c. is = n(n +1) 100 x101 Thus, 1+2+3+4+5,&c. continued to 100 terms, is. 2 =50x101=5050. Also the sum of any number of terms (n) of the series of odd numbers 1, 3, 5, 7, 9, 11, &c. is =n2. Thus, 1+3+5+7+9, &c. continued to 50 terms, is 502 = 2500. And if any three of the quantities, a, d, n, S, be given, the fourth may be found from the equation S= {2a+ (n-1)2}x, or (a+1)x 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 x 19=3+38=41, the last term. 20 20 3 Х =44 X =44X10=440, the sum 2 2 required. 20 Or, {2X3+(20–1)X2}*®=(6+19X2)X10=(6+ 38) X 10=44X10=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)=100-3X33=100-99=1, the last term. 34 And (100+1)x =101*** =101 X 17=1717, the 2 sum required. 34 Or, {2x100-(34-1) X 3}* (200— 33 X 3) 2 X 17=(200-99) X 17=101 X 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 to is to be used when the series is in. creasing, and the lower sign when it is decreasing; also the Last term laq+(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 trarel, wbo 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. OF GEOMETRICAL PROPORTION AND PROGRESSION. 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 geo. metrical 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. |