In order to make the preceding formula general, put p for 10, or the sum deposited, and n for the number of years, and we have

2. A man deposits 20 dollars annually, for 6 years, at 12 per cent. How much money is he entitled to at the end of the time?

3. Fifty persons are sent out annually for 10 years to settle a colony; the annual increase of those sent out is 3 per cent. How many persons will there be in the colony at the end of the time?

5. Supposing in the preceding formula, a, r and n are known, by what rule can you find the value of p ?

6. What sum must be deposited annually for 6 years, at 6 per cent., to amount to 560 dollars?

7. A man has a son 12 years of age; and he wishes to secure for him 1000 dollars when he shall be 21 years of age. There is a saving's bank that will allow 3 per cent. per annum on any sum deposited in it. How much must the man deposit annually until his son is 21 years old, to effect the object he has in view?

8. In the equation

pr (TM — 1)

-1

=a, what is the value of n?

Multiply by r1 and divide by pr, and we have

Taking the logarithm of both numbers, and we have

9. State the rule for finding the value of n, when a, r and p are known.

10. If 12 dollars be deposited annually, at 8 per cent., in what time will it amount to 180 dollars?

11. If 20 persons annually emigrate to a colony, and the rate of increase is 5 per cent., in how many years will the colony amount to 200 persons?

SECTION XCV.

Annuities.

1. What sum must a man deposit in bank, that he may draw out annually 10 dollars for 5 years, supposing that he is allowed 6 per cent. on the sum deposited?

This

Let x= the sum, and r = the amount of 1 dollar for 1 year. Then in one year, will amount to x X r. Hence xr will be the amount in bank at the end of the first year. He then draws out 10 dollars, and there remains xr- - 10. remains in bank the second year, and its amount is xr-10 times r, or xp2. - 10 r. Hence at the end of the second year, he is entitled to xr2-10 r. Out of this he draws 10 dollars, and there remains r2-10 r-10. This last sum remains in bank the third year, and amounts to x. 10 r2-10 r. Out of this he draws 10 dollars, and there remains x 3 -10r-10. This remains in bank the fourth year, and amounts to cr1 — 10 μ3 — 10 r2 — 10 r. Out of this he draws

10 μ2

10, and there remains xr-10 r3 — 10 r2 — 10 r-10. This remains in bank during the fifth year, and amounts to xr — 10 μa — 10 r3 — 10 r2-10 r. Out of this he draws 10, and there remains 5-10 — 10 r3 — 10 r2 10 r-10; and,

as the whole is now exhausted

x101

Or

In order to make this formula general, put a in place of 10. and n in place of 5, and we have

2. State the rule for finding x when a, r and n are known.

3. What sum must be deposited at 6 per cent., that the whole may be exhausted in 6 years by drawing 30 dollars per

annum ?

4. A man, by his will, directs that a sufficient sum shall be deposited in bank, at 4 per cent., to entitle his widow to draw out 50 dollars a year for 7 years. What sum must be deposited?

5. What is the value of a in the following equation:

6. State the rule for finding a when x, r and n are known.

7. If 150 dollars be deposited in bank, at 10 per cent., what sum may be drawn out annually, that the whole may be exhausted in 5 years?

8. From a colony of 500 persons, increasing at the rate of 5 per cent. per annum, how many must be taken away annually that there may be none left at the end of 7 years?

9. Find the value of n in the following equation:

10. State the rule for finding n, when a, r and x are known. 11. A man wishes to pay a debt of 500 dollars, on which he is to pay 6 per cent. per annum, in instalments of 50 dollars per annum. How long will it take him to pay the whole debt?

12. From a colony of 500 persons, increasing at the rate of 4 per cent. per annum, 60 persons are annually taken away. How long before there will be none left in the colony?

SECTION XCVI.

On Falling Bodies.

When a body is let fall through a space which offers no resistance, its motion increases constantly. During the first second, it passes through a space of 16 feet. Call this space g. At the end of the first second, the body has acquir ed a velocity which would carry it over a space 2 g, or 321 per second, if its velocity were not to increase during the next second of time. The motion, however, increases as much as it did during the first second, so that the body passes over a space 2 gg, or 3 g, during the next second; and has acquir

U

ed a velocity that would carry it over a space 4 g during the third second, if its motion were not to increase during the third second; but its velocity increases as much during the third second, as it did during the first; and, hence it passes over a space 4 g+g, or 5 g, during the third second.

Hence, it is evident that the falling body will pass over, during the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, seconds, the spaces g, 3 g, 5 g, 7 g, 9 g, 11 g, 13 g, 15 g, 17 g,

19 g.

It is evident that these spaces form a series in arithmetical progression, having g for the first term, and 2 g for the common difference.

1. What is the space passed over the 20th second?

Ans. 39 g. 2. What will represent the space passed over during the nth second? Ans. 2 (n-1) g+g, or 2 n g−g.

3. A body which has been falling for some time, is observed during the last second, to pass over a space of 3057 feet. How long has the body been falling?

Let

Then

n = the number of seconds,

2ng-g305,7

Or 324 n-16=3057

4. During what second, after a body commences falling, will it pass over a space of 15921 feet?

5. A falling body is observed to pass over a certain space in a second; if it had been falling 3 seconds longer, it would have passed over three times as much space in a second. How long had the body been falling?

6. The whole space passed over by a falling body, from the commencement of its motion, is evidently equal to the sum of the spaces passed over in each second. The space passed. over in 10 seconds, is 10 terms of the series, g, 3 g, 5 g, 7 g, 9 g, &c.

7. What is the sum of 10 terms of the series g, 3 g, 5 g, &c.? Ans. 100 g.

8. What will represent the space passed over by a body falling in 10 seconds? Ans. 100 g.