GEOMETRICAL PROGRESSION.

Any series of numbers, the terms of which increase by a common multiplier, or decrease by a common divisor, are said to be in geometrical progression; as 3, 6, 12, 24, 48; and 48, 24, 12, 6, 3.

The multiplier or divisor by which the series is increased or decreased is called the ratio.

The last term and sum of the series is found ry this

RULE.

1. Raise the ratio to the power whose index is one less than the number of terms given, which, being multiplied by the first term, will give the last term, or greater extreme.

2. Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder by ratio less one for the sum of the series.

EXAMPLES.

1. A thresher wrought 20 days, and received for the first day's labour 4 grains of wheat; for the second, 12; for the third, 36, &c. How much did his wages amount to, allowing 7680 grains to make a pint, and the whole to be disposed of at one dollar per bushel ?

Note. The first term in this question is 4, the ratio 3, the number of terms 20: therefore raise the ratio to the 19th power, which is one less than the number of terms.

Ratio 3, 9, 27, 81

81

81

648

6561 8th power of the ratio. 6561

6561

39366

32805

39366

43046721 16th power.
27 3d power.

301327047

86093442

1162261467 19th power.

4 1st term.

4649045868

3 Ratio.

13947137604

4 First term.

Ratio less one 2)13947137600

7680)6973568800 Sum of the series.

908016 pints, =14187 bushels.

14187 bushels, at 1 dol. p. bu. amount to 14187 dols. Ans.

2. Sold 24 yards of Holland, at 2 d. for the first yard, 4 d. for the second, 8 d. for the third, &c.; how much did it amount to? Ans. 139810 L. 2 s. 6 d.

3. Bought 30 bushels of wheat, at 2 d. for the first bushel, 4 d. for the second, 8 d. for third, &c.; what does the whole amount to, and what is the price per bushel on an average ?

Ans. (8947948 L. 19s. 6d. Amount.

298261 L. 12 s. 4 d. per bushel. 4. A merchant sold 30 yards of lace, at 2 pins for the first yard, 6 for the second, 18 for the third, &c., and disposed of the pins at 1000 for a farthing; how much did he receive for the lace? and how much did he gain by the sale, supposing the lace cost him 100 L. per yd.? Ans. Gained Received 214469929 L. 5 s. 34 d. 214466929 L. 5 s. 3 d.

5. A goldsmith sold 1 lb. of gold, at a farthing for the first ounce, a penny for the second, 4 d. for the third, &c., in quadruple proportion; how much did he receive for the whole, and how much did he gain by the sale, supposing he gave 4 L. per ounce for the gold?

Ans.

He received 5825 L. 8 s. 54 d. And gained 5777 L. 8 s. 54 d. 6. What sum would purchase a horse with 4 shoes, and eight nails in each shoe, at one farthing for the first nail, a halfpenny for the second, a penny for the third, &c. doubling to the last? Ans. 4473924 L. 5 s. 34 d.

7. A person married his daughter on new year's day, and gave her one dollar towards her portion, promising to double it on the first day of every month for one year; what was her portion? Ans. 4095 dols.

8. Suppose a man wrought 20 days, and received for the first day's labour 4 grains of corn, for the second 12, for the third 36, &c.; what did he receive for his labour, supposing 7680 grains to make a pint, and the whole to be sold at 2 s. 6 d. per bushel ?

Ans. 1773 L. 7 s. 6 d.

COMPOUND INTEREST,

BY DECIMALS.

The ratio in compound interest is the amount of one pound or dollar for one year; which is thus found:

As 100 1: 105 : 1.05 :

As 100: :: 105.5 : 1.055.

For quarterly amounts, take the 4th root of the ratio; for half yearly, the square root; and, for 3 quarters, the product of the quarterly and half yearly.

Thus, 1.03-1.007417; √✓1.03-1.014889; and 1.007417x1.014889-1.022416, for 3 quarters.

3

4

1.03 1.022416 1.014889 1.007417 .002500 31 1.035 1.026137 1.017349 1.008637 .002917 1.04 1.029852 1.019804 1.009853 .003333 41 1.045 1.033563 1.022252 1.011065 .003750 1.05 1.037270 1.024695 1.012272 .004167 51 1.055 1.040973 1.027132 1.013475 .004583 6 1.06 1.044671 1.029536 1.014674 .005000 6 1.065 1.048364 1.031988 1.015868 .005417 1.07 1.052053 1.034408 1.017058 .005833

5

19

The ratio involved to the time is the amount of 1 L. or dollar for the time given; as a square for 2 years, a cube for 3, &c. thus, 1.06 × 1.06×1.06×1.06.

or 1.06-1.262477-the 4th power of 1.06, or the ratio involved to 4 years.

When the ratio is to be involved to years and quarters, the power for the years is to be multiplied by the proper quarterly amount; as, 1.262477x1.044671= 1.318873 for 4 years, &c.

The power or the amount of 1 L. or dollar may also be obtained for months and days (nearly) by adding the monthly simple interest of 1L. or dollar, or proper parts thereof, to the amount of the quarter next preceding the expiration of the given time, for what that time exceeds the said quarter; thus,

=

Amount for yr. 1.029563: For 4 yrs.=1.318873
Int. of 1 L. for 1 mo.= .005000
One sixth for 5 dys.= .000833

.005000 .000833

For 7 mo. 5 dys. 1.035396: For 4 years, 10 mo. 5 dys.=1.324706

The ratio may be thus involved to any time whatever; but the operation is facilitated by the following tables; which may be extended to 100 years, or upwards, by multiplying the amount for 46, by that for the time above 46, &c.