EXAMPLES. 1. The extremes of an arithmetical feries are 3 and 39, and the number of terms 19; required the fam of the feries? 39 +3 33}}} Extremes. 2. It is required to find how many strokes the hammer of a clock would strike in a week, or 168 hours, provided it increased 1 at each hour? 3. Suppose a number of ftones were laid a yard dif tant from each other for the space of a mile, and the first, a yard from a basket; what length of ground will that man travel over who gathers them up fingly, returning with them one by one to the basket ? 3520+2 X 1760 2 = =3099360 yds.=1761 miles, Anf. N. B. In this question, there being 1760 yards in a mile, and the man returning with each stone to the basket, his travel will be doubled; therefore the first term will be 2, and the last 1760 × 2, and the number of terms 1760 4. A man bought 25 yards of linen in Arithmetical Progreffion; for the 4th yard he gave 12 fhillings, and for the last yard £3 15s.-What did the whole amount to, and, what did it average per yard? 75-12 22- I 3 the common difference, by which the first (term is found to be 3. Then, 75+3×25. 2 L48 15s. and the average price is (1. 19s. per yard. 5. Required the fum of the first 1000 numbers in their natural order? RULE. Divide the difference of the extremes by the common difference, and the quotient increased by I will be the number of terms required. EXAMPLES. Í. The extremes are 3 and 39, and the common difference 2; what is the number of terms? 2. A man, going a journey, travelled the first day 7 miles, the last day 51 miles, and each day increased his journey by 4 miles; How many days did he travel, and, how far? 51—7+1=12 days, and51+7×12 =348 miles, Anf. 4 2 PROBLEM 4 The extremes and common difference given, to find the fum of all the feries. RULE.-Multiply the fum of the extremes by their difference increased by the common difference, and the product divided by twice the common difference, will give the fum. I. EXAMPLES. If the extremes are 3 and 39, and the common difference 2; what is the fum of the feries? 39+3=42 Sum of the extremes. 39-3=36-Difference of extremes. 36+2=38 Difference of extremes increased by the (common difference. 2. A owes B a certain fum, to be discharged in a year, by paying 6d. the first week, 18d the fecond, and thus to increase every weekly payment by a fhilling, fhilling, till the laft payment be £2 11s. 6d. what is the debt? 5 1,5 +›5 × 5 1,5—5+ 1 = £67 125. Anf. IX 2 PROBLEM 5. The extremes and the fum of the feries given, to find the common difference. RULE. Divide the product of the fum and difference of the extremes, by the difference of twice the fum of the feries, and the fum of the extremes, and the quotient will be the common difference. EXAMPLE. Let the extremes be 3 and 39, and the fum 399: what is the common difference? 42 Sum of the extremes = =39+3= PROBLEM 6. The extremes and the fum of the feries given to find the number of terms. RULE. Twice the fum of the feries, divided by the fum of the extremes, will give the number of terms. EXAMPLE. EXAMPLE. Let the extremes be 3 and 39, and the fum of the series 399; what is the number of terms? Sum of the feries= 399 X 2 Sum of the extr. 39+3=42)798(19 the Anf. THEOREM I.-If four quantities, 2, 6, 4, 12, be in geometrical proportion, the product of the two means, 6 x 4, will be equal to that of the two extremes, 2 X 12, whether they are continued, or discontinued; and, if three quantities, 2, 4, 8, the fquare of the mean is equal to the product of the two extremes. THEOREM 2.-If four quantities, 2, 6, 4, 12, are fuch that the product of two of them, 2 X 12, is equal to the product of the other two, 6 X 4, then are thofe quantities proportional. THEOREM 3.-If four quantities, 2, 6, 4, 12, are proportional, the rectangle of the means, divided by either extreme, will give the other extreme. THEOREM 4-The products of the correfponding terms of two geometrical proptions are alfo proportional. That is, if 26::4: 12 and : 45: 10; then will 2 X 2: 6 × 4:: 4 X 5: 12 X 10. THEOREM *And if any quantities be proportional, their squares, bes, &c. will likewise be proportional, |