EXAMPLES 1. Find the 8th term of the series 2, 4, 8, 16, SOLUTION: - 1 l= am ; .. = 2 × 27 = 256, ans. 2. Find the first term of the series. . . . 48, 96, 192, having 6 terms. -1 SOLUTION: 1= ar2- ; .. 192 = a × 25; a = 6, ans. 3. Given a = 6, == 1536, r = 4; find n. SOLUTION: 1=ar"-1; .. 1536 = 6 x 4"-1. 256 4n-1. = 4"-1; .. n−1=4 n = 5, ans. ;. S= 6-1 7775, ans. Exercise XXIII 1. Find the 11th term of the series 2, 6, 18, 54, 2. Find the 8th term of the series 3, 32, 33, 34, NOTE. In a descending series with an infinite number of terms, the last term is 0, and the formula is written 5. Find the sum of 4, 2, 1, 1, etc. to infinity. 6. Insert 4 geometric means between 6 and 192. 7. Write a series of 4 terms, whose 1st term is 4 and whose ratio is 1. 8. If a frog should jump 10 feet the first jump, 5 feet the second, 2.5 feet the third, etc. forever, how far would it go? 9. If a child should receive one cent at birth, 2 cents on the second birthday, 4 cents on the third, etc., how much would he be worth when 21 years of age? NOTE.-Problems in compound interest may be solved by the principles of geometrical progression. The terms are as follows: απ principal. r = 1 + rate per cent for one interval. n = number of intervals + 1. 7 = amount. 10. Find the amount of $250 for 4 years, at 6% per annum compound interest. SOLUTION: 1 arn -1; = 7 = $250 × 1.064 $315.619 +, ans. 11. Find the amount of $400 for 3 years, at 6% per annum compound interest. 12. Find the compound interest of $800 for 5 years, at 5%. 13. What principal will amount to $322.51 in 24 years, at 5% per annum compound interest? 14. In how many years will $80 amount to $106.48, at 10% compound interest? 15. At what rate per cent compound interest will $750 amount to $946.86, in four years? NOTE. Circulating decimals may be changed to common fractions by using the principles of geometrical progression. Since a circulate always forms an infinite decreasing series, this formula is used: α S= 1 r 16. Change 0.27 to an equivalent common fraction. (1) 0.27 = 0.27 +0.0027 + 0.000027 + (2) Given a = 0.27, r = 0.01, n = infinity, to find s. CHAPTER VIII METHOD OF ATTACK 205. Often pupils become discouraged and fail in arithmetic, because they neither understand the real nature of mathematical problems nor have any definite method of attacking them. While under each subject model solutions are given for problems arising under that special head, it is thought wise to give in this chapter some suggestions and examples which will be helpful in the solution of problems in general. 206. The Nature of Problems.-A problem consists of at least two parts: a known part and an unknown part. 1. The known part is a statement of conditions or relations forming a basis from which the unknown part may be determined. 2. The unknown part is frequently in the form of a question and is always the part required. Problem: If 8 tops cost 40c., what will 17 tops cost? 1. Known: 8 tops cost 40c. PARTS: { 2. Unknown: What will 17 tops cost? Sometimes the known part is not expressed, but implied, as in the following: How many quarts in 10 gallons? The known part is, there are 4 quarts in 1 gallon. NOTE.-Before one attempts to solve a problem, he should determine the two parts and see clearly that a proper relation exists between them. 207. Classification.-Problems of one basis, considered with reference to the relation of the given part to the required part, are divided by Prof. J. A. Ferrell into four classes, as follows: Class I. Given a number, to find (1) a part of it, or (2) a multiple of it; as, 1. What is the value of of an article which cost $7.20? 2. What will 50 books cost, at $2.50 each? Class II. Given a part of a number, to find (1) the number, (2) a multiple of it, or (3) a part of it; as, 1. 20% of a sale is $540. What was the whole amount of the sale? 2. of a certain farm is 25 acres. 3 such farms? How many acres in 3. of a certain article is worth $.75. What is of it worth? Class III. Given a multiple of a number, to find (1) the number, (2) a part of the number, or (3) another multiple of it; as, 1. If 35 hats sell for $105, find the price of 1 hat. 2. At the rate of $120 in 2 months, how much money can be earned in of a month? 3. 12 pounds Troy equal 144 ounces; how many ounces do 7 pounds Troy equal? Class IV. Given two numbers, to find (1) what part one is of the other, or (2) what multiple one is of the other; as, 1. A commission merchant charges $96 for making a sale of $3200. What per cent does he charge? |