Higher Algebra

8. Insert 9 arithmetical means between and 12.

9. Insert 5 arithmetical means between 17% and 14. 10. Insert 20 arithmetical means between 16 and 26.

11. Insert 6 arithmetical means between a + b and 8 a – 13 b.

Problems.

13. Pr. 1. The sum of four numbers in arithmetical progression is 16, and their product is 105. What are the numbers?

We can express the four required numbers in terms of two unknown numbers.

Let x 3d, x d, x+d, x+3d be the four required numbers. Then, by the first condition,

whence

(x − 3 d) + (x − d) + (x + d) + (x + 3 d) = 16;

By the second condition,

x = 4.

(x − 3 d) (x − d) (x + d) (x + 3 d) = 105,

(x2 — 9 d2) (x2 — d2) = 105.

Substituting 4 for x and reducing, 9 d 160 d2 = - 151.
From this equation we obtain d±1, and ± √151.
The corresponding numbers are,

when d1: 1, 3, 5, 7; when d=-1: 7, 5, 3, 1;

when d = √151:

4-√151, 4 - {√/151, 4+}√151, 4 +√151;

when d√/151:

4+√151, 4+{√√/151, 4 − {√/151, 4 — √151. Notice the advantage of assuming the required numbers as in the above example. Had we assumed x, x+d, x+2d,

3d as the required numbers, the solution would have involved an equation of the fourth degree which could not have been solved as a quadratic.

Pr. 2. Find the sum of all the numbers of three digits which are multiples of 7.

The numbers of three digits which are multiples of 7 are

7 x 15, 7 × 16, 7 x 17, ..., 7 x 142.

The series within the parentheses is an arithmetical progression, in which a1 = 15, d= = 1, n = 128, and a128

142.

The required sum is therefore 7 × 10,048, = 70,336.

EXERCISES III.

1. Find the sixth term, and the sum of eleven terms, of an A. P. whose eighth term is 11 and whose fourth term is - 1.

2. The sixteenth term of an A. P. is 5, and the forty-first term is 45. What is the first term, and the sum of twenty terms?

3. Find the sum of all the even numbers from 2 to 50 inclusive.

4. Find the sum of thirty consecutive odd numbers, of which the last is 127.

5. The sum of the eighth and fourth terms of an A. P. of twenty terms is 24, and the sum of the fifteenth and nineteenth terms is 68. What are the elements of the progression ?

6. The product of the first and fifth terms of an A. P. of ten terms is 85, and the product of the first and third terms is 55. What are the elements of the progression ?

7. The first term of an A. P. of thirty terms is 100, and the sum of the first six terms is five times the sum of the six following terms. What are the elements of the progression ?

8. The sum of the second and twentieth terms of an A. P. is 10, and their product is 2311. What is the sum of sixteen terms?

9. The sixth term of an A. P. is 30, and the sum of the first thirteen terms is 455. What is the sum of the first thirty terms?

10. What value of x will make the arithmetical mean between x and x equal to 6 ?

11. Find the sum of all even numbers of two digits.

12. How many consecutive odd numbers beginning with 7 must be taken to give a sum 775?

13. Insert between 0 and 6 a number of arithmetical means so that the sum of the terms of the resulting A. P. shall be 39.

14. Find the number of arithmetical means between 1 and 19, if the first mean is to the last mean as 1 to 7.

15. The sum of the terms of an A. P. of six terms is 66, and the sum of the squares of the terms is 1006. What are the elements of the progression ?

16. The sum of the terms of an A. P. of twelve terms is 354, and the sum of the even terms is to the sum of. the odd terms as 32 to 27. What is the common difference?

17. How many positive integers of three digits are there which are divisible by 9? Find their sum.

18. If the sum of m terms of an A. P. is n, and the sum of n terms is m, what is the sum of m + n terms? Of m - n terms?

19. Show that the sum of 2 n + 1 consecutive integers is divisible by 2 n + 1.

20. Prove that if the same number be added to each term of an A. P., the resulting series will be an A. P.

21. Prove that if each term of an A. P. be multiplied by the same number, the resulting series will be an A. P.

22. Prove that if in the equation y: = ax + b, we substitute c, c + d, c+2d, ..., in turn for x, the resulting values of y will form an A. P. 23. Prove that if a2, b2, c2 form an A. P., then

24. If a, b, c form an A. P., then

} (a + b + c)3 = a2 (b + c) + b2 (a + c) + c2 (a + b).

25. A laborer agreed to dig a well on the following conditions: for the first yard he was to receive $2, for the second $2.50, for the third $3, and so on. If he received $42.50 for his work, how deep was the well? 26. On a certain day the temperature rose hourly from 5 to 11 A.M., and the average temperature for that period was 8°. What was the temperature at 8 A.M.?

27. Twenty-five trees are planted in a straight line at intervals of 5 feet. To water them, the gardener must bring water for each tree separately from a well which is 10 feet from the first tree and in line with the trees. How far has the gardener walked when he has watered all the trees?

28. Two bodies, A and B, start at the same time from two points, C and D, which are 75 feet apart, and move in the same direction. A moves 1 foot the first second, 3 feet the second, 5 feet the third, etc.; B moves 3 feet the first second, 4 feet the second, 5 feet the third, etc. How long will it take A to overtake B?

29. A number of equal balls are placed in the form of a solid equilateral triangle in the following way: one ball is placed at the vertex, under this are placed two balls, under these two are placed three balls, and so on. If the number of balls is increased by 4, they can be placed in the form of a solid rectangle whose base is equal to the base of the triangle, and whose altitude is 3 balls shorter than the base. How many balls are in the triangle ?

§ 3. GEOMETRICAL PROGRESSION.

1. A Geometrical Series, or as it is more commonly called a Geometrical Progression (G. P.), is a series in which each term after the first is formed by multiplying the preceding term by a constant number. See § 1, Art. 1, (2).

Evidently this definition is equivalent to the statement that the ratio of any term to the preceding is constant.

For this reason the constant multiplier of the first definition is called the Ratio of the progression.

Let

and

a stand for the first term of the series,

a, for the nth (any) term,

r for the ratio,

S, for the sum of n terms.

The five numbers a1, an, r, Sn, n, are called the Elements of the progression.

2. The ratio may be either larger or smaller than 1; in the former case the progression is called a rising or ascending progression; in the latter a falling or descending progression.

If the terms are all positive, the words increasing and decreas ing may be used for ascending and descending, respectively.

3. In a geometrical progression any term is equal to the first term multiplied by a power of the ratio whose exponent is one less than the number of the required term, i.e.,

By the definition of a geometrical progression

а1 = а1, а2 = а ̧Ñ‚ ɑ ̧= α2r = a ̧r2, α = α ̧r = α,3, etc.

(I.)

The law expressed by the relations for these first four terms is evidently general, and since the exponent of r is one less than the number of the corresponding term, we have

This relation may also be used to find not only a when a r, and n are given, but also to find the value of any one of the four numbers when the other three are given.

Ex. 2. If a1 = 4, a ̧ = }, n = 6, then | = 4 r3, whence r = It is important to notice that, while a, a, and r may be positive or negative, integral or fractional, n must be a positive integer. Consequently a, a,, r cannot be assumed arbitrarily. As yet the value of n can be determined from (I.) only by inspection.