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, &c. then will 2 + 104+8=2×6=12.

And, if the series be a, a+d, a + 2d, à + 3d, a+4d, &c. then will a+ (a + 4d) = a + d + (a+ 3d) = 2 × (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, a +d, a + 2d, a+3d, 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 − (n − 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+

And the sum of a + (a + d) + (a + 2d) + (a + 3d), &c.

n

+l, is = (a + 1) × where is the

2'

last term, and n the number of terms.

Or, the sum of any increasing arithmetical series may be found, 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 terms,

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 before.

n

2

Thus, the sum of a+ (a + d) + (a + 2d) + (a + 3d) + (a + 4d), &c. to n terms, is = 2a + (n − 1)d × And the sum of a+ (a−d) + (a − 2d) + (a 3d) + (a− 4d), &c. to n terms, is = 2a- (n − 1)d × 2 (i).

EXAMPLES.

1. The first term of an increasing arithmetical series is 3, the common difference 2, and the numof terms 20; required the sum of the series.

First, 3+2(20-1)=3+2 × 19=3+38=41, the last term.

And (3+41) ×20=44 ×

20

=44 × 10=440, the

2

sum required.

(i) The sum of any number of terms (n) of the series of natural

numbers 1, 2, 3, 4, 5, 6, 7, &c. is

Thus, ì

100 × 101

2

=

n(n + 1)

2

+ 2 + 3 4+5, &c. continued to 100 terms, is

+

50 x 1015050.

Also, the sum of any number of terms (n) of the series of odd numbers 1, 3, 5, 7, 9, 11, &c. is =n2.

And if any three of the quantities a, d, n, s, be given, the fourth may be found from the equation

n

s = {2a ±(n − 1)d} × 2, or (a + 1)×2

Where the upper sign is to be used when the series is increasing, and the lower sign. - when it is decreasing; also the last term la + (n-1)d, as above.

Or, (2 × 3 + 20 — 1 × 2) × 20 = (6+ 19 × 2) × 10=

X

(6+38) × 10=44 x 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 x 33 100991, the last term.

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 the compass of a day? Ans. 300.

6. 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.

7. 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.

(M) GEOMETRICAL PROPORTION is that relation of two quantities of the same kind, which arises from considering what part, or parts, the one is of the other, or how often it is contained in it.

When four quantities are compared together, the first and third are called the antecedents, and the second and fourth the consequents.

Ratio is the quotient, which arises from dividing the antecedent by the consequent, or the consequent by the antecedent; observing 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.

1 1 9' 3'

1 1

are

Thus, 2, 6, 9, 3, and a, ar, br, b, are inversely proportional; because 2, 6, directly proportional.

and a, ar, br b'

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, ars, art, &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 = 42 and a × 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 =√36=6.

And a geometric mean between a and b is =√ab(k).

(k) If two, or more, geometrical means between any two quan.