8. Two persons, A and B, have both the same income; A saves one fifth of his every year; but B, by spending 150 dollars per annum more than A, at the end of 8 years finds himself 400 dollars in debt: What is their income, and what does each spend per annum ? Their income is 500 dollars per annum. A spends 400 dols., and B 550. Ans. S ARITHMETICAL PROGRESSION. Any rank or series of numbers, increasing or decreasing by a common difference, is said to be in arithmetical progression; as 2, 4, 6, 8, 10, and 6, 5, 4, 3, 2, 1. The numbers which form the series are called the terms. The first and last terms are called the ex tremes. Note.-In any series of numbers in Arithmetical Progression, the sum of the two extremes is equal to the sum of any two terms equally distant from them; as in the latter of the above series 6+14+3, and 5+2. When the number of terms is odd, the double of the middle term is equal to the sum of the two extremes, or any two terms equally distant from the middle term; as in the former of the foregoing series 6x2=2+10, and 4+8. CASE 1. The first term, common difference, and number of terms given, to find the last term, and sum of all the terms. RULE. 1. Multiply the number of terms, less 1, by the common difference, and to the product add the first term, the sum is the last term. 2. Multiply the sum of the two extremes by the number of terms, and half the product will be the sum of all the terms. EXAMPLES. 1. The first term of a certain series in arithmetical progression is 2, the common difference is 2, and the number of terms 15; what is the last term, and the sum 2. Bought 15 yards of linen, at 2 cents for the first yard, 4 for the second, 6 for the third, &c., increasing 2 cents every yard; what was the cost of the last yard, and what was the cost of the whole? Ans. The last yard cost 30 cts., the whole cost $2.40. 3. Sold 20 yards of silk, at 3d. for the first yard, 6 d. for the second, 9 d. for the third, &c., increasing 3d. every yard; what sum did it amount to? Ans. 2 L. 12 s. 6 d. 4. Sixteen persons gave charity to a poor man; the first gave 5 d., the second 9 d., and so on in an arithmetical progression; how much did the last person give, and what sum did the man receive? Ans. The last gave 5s. 5 d., sum received, 2 L. 6 s. 8 d. 5. If 100 stones be laid two yards distant from each other in a right line, and a basket placed two yards from the first stone; what distance must a person travel to gather them singly into the basket? Ans. 11 miles, 3 fur. 180 yds. 6. A merchant sold 1000 yards of linen, at 2 pins for the first yard, 4 for the second, and 6 for the third, &c. increasing two pins every yard; how much did the linen produce, when the pins were afterwards sold at 12 for a farthing? Ans. 86 L. 17 s. 10 d. CASE 2. When the two extremes and number of terms are given, to find the common difference. RULE. Divide the difference of the extremes by the number of terms, less one; the quotient will be the common difference. EXAMPLES. 1. Twenty and sixty are the two extremes of a certain series in arithmetical progression, and 21 is the number of terms; what is the common difference? Ans. 2. 60 20 }} extremes. 21-1-20)40 Difference of extremes. 2 Common difference. 2. There are 21 men whose ages are equally distant from each other in arithmetical progression; the youngest is 20 years old, and the eldest 60; what is the common difference of their ages, and the age of each man? Ans. Common difference 2 years 60 years is the age of the first man. age of the second. age of the third, &c. 60-2=58 3. A debt is to be paid at 16 different payments in arithmetical progression; the first payment to be 14 L. and the last 100 L.: what is the common difference, each payment, and the whole debt? (Common difference 5 L. 14 s. 8 d. Ans. First payment 14 L. Second, 19 L. 14 s. 8 d. Third, 25 L. 9 s. 4 d. &c. 4. A person is to travel from Philadelphia to a certain place in 16 days, and to go but 4 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 79 miles; what is the common difference; and what the whole distance? Ans. (Common difference 5 miles. Distance 664 miles. GEOMETRICAL PROGRESSION. Any series of numbers, the terms of which increase by a common multiplier, or decrease by a common divisor, are said to be in geometrical progression; as 3, 6, 12, 24, 48; and 48, 24, 12, 6, 3. The multiplier or divisor by which the series is increased or decreased is called the ratio. The last term and sum of the series is found by this RULE. 1. Raise the ratio to the power whose index is one less than the number of terms given, which, being multiplied by the first term, will give the last term, or greater extreme. 2. Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder by ratio less one for the sum of the series. EXAMPLES. 1. A thresher wrought 20 days, and received for the first day's labour 4 grains of wheat; for the second, 12; for the third, 36, &c. How much did his wages amount to, allowing 7680 grains to make a pint, and the whole to be disposed of at one dollar per bushel ? Note. The first term in this question is 4, the ratio 3, the number of terms 20: therefore raise the ratio to the 19th power, which is one less than the number of terms. |