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1+r:1=1: present value of 1st payment.

1

1+r'

1

1+r: 1=1+r' (1+r)2' present value of 2d payment.

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1+r:1= (1+r)2 ' (1+r)3?

present value of 3d payment.

1+r:1= (1+r)* (1+r)' present value of 4th payment.

1+r:1=

1

1

(1+r)”−1 ° (1+r)”›

present value of nth payment.

124. The present value of an annuity of L.1 for n years is therefore the sum of the series.

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This is evidently a geometrical series, in which the first

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ing its sum as in geometrical progression, and putting p for the sum that is the present value of the annuity, we have

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1

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1+r (1+r)*

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(2.)

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125. If the annuity is to continue for ever, then ʼn be

comes infinite, and also (1+r)"; hence

1

(1+r)"

may be con

sidered as 0, and therefore we have for the present value of an annuity of L.1, payable for ever,

p=-, value of a perpetuity of L.1.

It is plain, that if the annuity be a pounds instead of one, it will just be a times as great as before; and therefore the present value of an annuity of a pounds, payable for n years, will be

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and that of a perpetuity of a pounds will be

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126. When an annuity is only to commence n years hence, and then continue for t years, it is called a deferred annuity, and it is plain that its present value will be the difference between the present value of an annuity to continue for t+n years, and another to continue for n years; but we have seen (5) that the present value of an annuity of L.1 to continue t+n years, is (1+r)t+n_1 and that the present value of an annuity to continue n years is

(1+r)n-1.

r(1+r)t+n

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; the difference of these expressions is therefore r(1+r)n the present value of an annuity commencing n years hence and continuing afterwards for t years; reducing these expressions to a common denominator and subtracting, we (1+r)t-1 and therefore the present value of an anr(1+r)t +n' nuity to commence n years hence, and afterwards to con(1+r)*-1

have

tinue for t years, is p=(1+r)+ and if the annuity be a pounds per annum, instead of one, it is plain that the whole result will be a times as great; therefore the present value of an annuity of a pounds per annum deferred for n years, and then payable t years, is p_a{(1+r)—1}

r(1+r)e+n

If the annuity be payable for ever after n years, then t, and consequently (1+r) become infinite, dividing both numerator and denominator of the above expression by

1

'(1+r), and observing that becomes 0, we have for (1+r)t the present value of a perpetuity of a pounds deferred for

a

n years, p= (1+r)n'

127. To find the amount of an annuity left unpaid any number of years, at compound interest. Let A be the annuity, then the amount of the first payment which is foreborne for n-1 years will be A(1+r); of the second for n-2 years will be 4(1+r)"; &c.

the whole

n terms;

amount=A(1+(1+r)+(1+r)2+, &c.) to

or the amount= · 4 ((1+r)*−1 ). Art. 96. (3.)

Ex. 1. What is the present value of a pension of L.100, payable yearly, for 20 years, at 5 per cent. compound interest? Ans. L.1246, 4s. 51d.

2. What is the present value of a perpetual annuity of L.100, payable yearly, interest at 5 per cent.? Ans. L.2000.

3. What is the present value of an annuity of L.100, payable half-yearly for 20 years, interest at 5 per cent. per annum, also payable half-yearly? Ans. L.1255, 2s. 101d.

4. What is the present value of a perpetuity of L.100 per annum, payable half-yearly, interest at 5 per cent. per annum, being also payable half-yearly? Ans. L.2000. 5. What is the present value of an annuity of L.100 to commence 10 years hence, and then continue for 30 years, interest at 4 per cent.? Ans. L.1168, 3s. 71d.

6. What is the present value of an annuity of L.50, to commence 8 years hence, and then to continue for 42 years, interest at 5 per cent.? Ans. L.589, 12s. 8d.

per cent. ?

7. In what time will a pension of L.50 amount to L.1000, interest at 5 Ans. 14.2 years. 8. To what sum will an annuity of L.24 amount in 20 years, when improved at 5 per cent.?

Ans. L.793, 11s. 8d.

PROMISCUOUS EXERCISES.

1. It is required to divide each of the numbers 11 and 17 into two parts, so that the product of the first parts of each may be 45, and of the second 48. Ans. 5, 6, and 9, 8.

2. Divide each of the numbers 21 and 30 into two parts, so that the first part of 21 may be three times as great as the first part of 30, and that the sum of the squares of the remaining parts may be 585. Ans. 18, 3, and 6, 24. 3. A gentleman left L.210 to three servants, to be divided in continued proportion, so that the first shall have L.90 more than the last : find their legacies.

Ans. L.120, L.60, and L.30. 4. There are two numbers, whose product is 45, and the difference of their squares is to the square of their difference as 7 is to 2: what are the numbers? Ans. 9 and 5.

5. A and B engage in partnership with a capital of L.100: A leaves his money in the partnership for 3 months, and B for 2 months, and each takes out L.99 of capital and profit: determine the original contribution of each.

Ans. A L.45, and B L.55.

PLANE GEOMETRY.

GEOMETRY is that branch of Mathematics which treats of the properties of measurable magnitudes.

Magnitudes are of three kinds, viz. lines having length only, surfaces having length and breadth, and solids having length, breadth, and thickness.

That branch of Geometry which treats of lines and surfaces is called Plane Geometry, and that which treats of the properties of solid bodies is called Solid Geometry.

DEFINITIONS.

1. A point is that which has position but no magnitude. 2. A line is length without breadth.

3. The extremities of a line are points.

4. A straight line is that which lies evenly between its extreme points.

5. A superficies is that which hath only length and breadth.

6. The extremities of a superficies are lines.

7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

8. A plane rectilineal angle is the inclination of two straight lines which meet in a point, but are not in the same straight line.

NOTE. When there A are several angles at one point, as at B, each of the angles must be named by three letters, and the letter at the angular point must be placed between other two; thus, the

the B

D

E

C

angle formed by the lines AB and BD meeting in the point B, is called the angle ABD or DBA; also the angle formed by the straight lines DB and BC, is called the angle DBC or CBD; but when there is only one angle at the point, as at E, it may be called simply the angle at E.

9. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

10. An obtuse angle is that which is greater than a right angle.

11. An acute angle is that which is less than a right angle.

12. A term or boundary is the extremity of any thing. 13. A figure is that which is enclosed by one or more boundaries.

14. A circle is a plain figure bounded by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within it to the circumference are equal to one another;

15. And this point is called the centre.

16. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

17. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

18. A straight line drawn from the centre to the circumference of a circle is called a radius.

19. A straight line which is terminated both ways by the circumference, but does not pass through the centre, is called a chord.

20. The part of the circumference cut off by the chord is called an arc.

21. The figure bounded by the chord and arc is called a segment.

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22. Rectilineal figures are those which are contained by straight lines.

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