769. 1. The first term of a geometrical progression is 3, the ratio 4; what is the 8th term?

2. The first term of a geometrical progression is 1, and the ratio 2; what is the 12th term?

3. The extremes are 4 and 2916, and the ratio 3; what is the number of terms?

4. The extremes of a geometrical progression are 2 and 1458, and the ratio 3; what is the sum of all the terms?

5. The first term is 3, the seventeenth 196608; what is the sum of all the terms?

6. A man traveled 6 days; the first day he went 5 miles and doubled the distance each day; his last day's ride was 260 miles; how far did he travel?

7. Supposing an engine should start at a speed of 3 miles. an hour, and the speed could be doubled each hour until it equalled 96 miles, how far would it have moved in all, and how many hours would it be in motion?

8. The first term of a geometrical progression is 4, the 7th term is 2916; what is the ratio and the sum of the series?

ANNUITIES.

770. An Annuity is a fixed sum of money, payable annually, or at the end of any equal periods of time.

771. The Amount or Final Value of annuity is the sum of all the payments, each payment being increased by its interest from the time it is due until the annuity ceases.

2. The Present Worth of an annuity is such a sum of money as will amount, at the given rate per cent, in the given time, to the Amount or Final Value of the annuity.

773. An Annuity at Simple Interest forms an arithmetical progression whose common difference is the interest on the given annuity for one interval of time.

Thus an annuity of $400 for 4 years, at 7% simple interest, gives the following progression :

Observe, there is no interest on the last payment; hence it forms the 1st Term. The payment before the last bears one year's interest, hence forms the 2d Term; and so on with the other terms.

Hence all problems in annuities at simple interest are solved by arithmetical progression.

774. An Annuity at Compound Interest forms a geometrical progression whose common multiplier is represented by the amount of $1 for one interval of time.

Thus an annuity of $300 for 4 years, at 6% compound interest, gives the following progression :

Observe carefully the following:

(1.) The last payment bears no interest, and hence forms the 1st Term of the progression.

(2.) The payment before the last, when not paid until the annuity ceases, bears interest for one year; hence its amount is $300 × 1.06 and forms the 2d Term.

(3.) The second payment before the last, bears interest when the annuity ceases, for two years; hence its amount at compound interest is $300 × 1.06, the amount for one year, multiplied by 1.06, equal $300 × 1.06 × 1.06, and forms the 3d Term, and so on with other terms.

Hence all problems in annuities at compound interest are solved by geometrical progression.

75. 1. What is the amount of an annuity of $200 for 6 years at 7% simple interest?

2. A father deposits $150 annually for the benefit of his son, beginning with his 12th birthday; what will be the amount of the annuity on his 21st birthday, allowing simple interest at 6%?

3. What is the present worth of an annuity of $600 for 5 years at 8%, simple interest?

4. What is the amount of an annuity of $400 for 4 years at 7%, compound interest?

5. What is the present worth of an annuity of $100 for 6 years at 6%, compound interest?

6. What is the present worth of an annuity of $700 at 8%, simple interest, for 10 years?

7. What is the amount of an annuity of $500 at 7%, compound interest, for 12 years?

8. What is the present worth of an annuity of $350 for 9 years at 6%, compound interest?

9. At what rate % will $100 amount to $119.1016 in 3 years, at compound interest ?

This example and the four following should be solved by applying the formulæ for geometrical progression on page 337.

10. At what rate % will $1000 amount to $1500.73 in 6 years, compound interest?

11. The amount of a certain sum of money for 12 years, at 7% compound interest, was $1126.096; what was the original sum?

12. What sum at compound interest 8 years, at 7%, will amount to $4295.465?

13. In how many years will $20 amount to $23.82032, at 6% compound interest?

6. A Line is that which has only length.

7. A Straight Line is a line which has the same direction at every point.

8. A Curved Line is a line which changes its direction at every point

9. Parallel Lines are lines which have the same direction. 780. An Angle is the opening between two lines which meet in a common point, called the vertex.

781. When a line meets another line, making, as shown in (1), two equal angles, each angle is a Right Angle, and the lines are said to be perpendicular to each other.

782. An Obtuse Angle, as shown in (3), is greater than a right angle, and an Acute Angle, as shown in (4), is less than a right angle. Angles are read by using letters, the letter at the vertex being always read in the middle. Thus, in (2), we read, the angle BAC or CAB.

783. A Plane is a surface such that if any two points in it be joined by a straight line, every point of that line will be in the surface.

784. A Plane Figure is a plane bounded either by straight or curved lines, or by one curved line.

785. A Polygon is a plane figure bounded by straight lines. It is named by the number of sides in its boundary; thus:

Trigon.

Tetragon.

Pentagon.

Hexagon, and so on.

Observe, that a regular polygon is one that has all its sides and all its angles equal, and that the Base of a polygon is the side on which it stands.

786. A Trigon is a three-sided polygon. It is usually called a Triangle on account of having three angles.

Observe, a right-angled triangle has ONE right angle, an acute-angled triangle has THREE acute angles, and an obtuse-angled triangle has ONE obtuse angle.

Observe, also, as shown in (2) and (3), that the Altitude of a triangle is the perpendicular distance from one of its angles to the side opposite.

787. An Equilateral Triangle is a triangle whose three sides are equal.

788. An Isosceles Triangle has two of its sides equal.

789. A Scalene Triangle has all of its sides unequal.

790. A Tetragon is a four-sided polygon, It is usually called a Quadrilateral.