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the number of terms. Or, the sum of any increasing arithmetical series may be found, without considering the last term, by adding the product of the common difference by the number of terms less one to twice the first term, and then multiplying the result by half the number of teruns. 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--(a-H d)+(a+2d)+(a+3d) + (a + 4d), &c. continued to n terms, we shall have

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(y) The sum of any number of terms (n) of the series of natural numbers 1, 2,3,4, 5, 6, 7, &c. is - refu

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And if any three of the quantities, a, d t wnay be found from the equation a, d, n, S, be given, the fourth

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× 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 7 - Ans. 300.

where the upper sign + is to be used when the series is oncreasing, and the lower sign — when it is decreasing; also the last term l=a+(n-1) d, as above, o

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 10, the common difference ; and the number of terms 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.

OF
GEOMETRICAL PROPORTION

AND

PROGRESSION. y

GEeMETRICAL 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, ar”, 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

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... 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, ar”, ar”, ar", &c. are series in geometrical progression. The most useful properties of geometrical proportion and progression are contained in the following theorems : 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 × 8–4°, and a X c=b2. 2. Hence, a geometrical mean proportional, between any two quantities, is equal to the square root of their product. Thus, a geometric mean between 4 and 9 is = v36 =6. And a geometric mean between a and b is = vab.

3. If four quantities be in geometrical proportion, the product of the two extremes will be equal to that of the means.

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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.

Thus, if the proportionals be 3, 9, 5, 15, or a, b, c,

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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.

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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.

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