2. How much water at 0 per gallon, must be mixed with wine at 90 cents per gallon, so as to fill a vessel of 100 gallons, which may be afforded at 60 cents per gallon? Ans. 334 gals. water, and 663 gals. wine. 3. A grocer having sugars at 8 cts. 16 cts. and 24 cts. per pound, would make a composition of 240 lb. worth 20 cts. per lb. without gain or loss; what quantity of each must be taken ? Ans. 40 lb. at 8 cts. 40 at 16 cts. and 160 at 24 cts. 4. A goldsmith had two sorts of silver bullion, one of 10 oz. and the other of 5 oz. fine, and has a mind to mix a pound of it so that it shall be 8 oz fine; how much of each sort must he take? Ans. 4 of 5 oz. fine, and 7 of 10 oz. fine. 5. Brandy at Ss. 6d. and 5s. 9d. per gallon, is to be mixed, so that a hhd. of 63 gallons may be sold for 12 128.; how many gallons must be taken of each? Ans. 14 gals. at 5s. 9d. and 49 gals. at $s. 6d. ARITHMETICAL PROGRESSION. ANY rank of numbers more than two, increasing by common excess, or decreasing by common difference, is Isaid to be in Arithmetical Progression. So 52, 4, 6, 8, &c. is an ascending arithmetical series: 18, 6, 4, 2, &c. is a descending arithmetical series: The numbers which form the series, are called the terms of the progression; the first and last terms of which are called the extremes.* PROBLEM I. The first term, the last term, and the number of terms being given, to find the sun of all the terms. *A series in progression includes five parts, viz. the first term, last term, number of terms, common difference, and sum of the series. By having any three of these parts given, the other two may be found, which admits of a variety of Problems; but most of them are best understood by an algebraic process, and are here omitted. RULE. Multiply the sum of the extremes by the num terms, and half the product will be the answer. EXAMPLES. 1. The first term of an arithmetical series is 3, th term 25, and the number of terms 11; required the of the series. 23+3 26 sum of the extremes. 2. How many strokes does the hammer of a erike, in twelve hours? Ans. 3. A merchant sold 100 yards of cloth, viz. the yard for 1 ct. the second for 2 cts. the third for S cts I demand what the cloth came to at that rate? Ans. $50 4. A man bought 19 yards of linen in arithmetical gression, for the first yard he gave 18. and for the las Il. 17s. what did the whole come to ? Ans. £18 1 5. A draper sold 100 yards of broadcloth, at 5 cts the first yard, 10 cts. for the second, 15 for the third, increasing 5 cents for every yard; what did the w amount to, and what did it average per yard? Ans. Amount $252, and the average price is $2, 55 5 mills per yard. 6. Suppose 144 oranges were laid 2 yards distant f each other, in a right line, and a basket placed two ya from the first orange, what length of ground will that travel over, who gathers them up singly, returning them one by one to the basket? Ans. 25 miles, 5 furlongs, 180 yd. PROBLEM II. The first term, the last term, and the number of te given, to find the common difference. RULE. Divide the difference of the extremes by the num of terms less 1, and the quotient will be the common ference. EXAMPLES 1. The extremes are 3 and 20, and the number of terms 14, what is the common difference? 29 2 Extremes. Number of terms less 1-13)26(2 Ans. 2. A man had 9 sons, whose several ages differed alike, the youngest was 3 years old, and the oldest 55; what was the common difference of their ages? Ans. 4 years 3. A man is to travel from New-London to a certain place in 9 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 43 miles: Required the daily in crease, and the length of the whole journey? Ans. The daily increase is 5, and the whole journey 207 miles. 4. A debt is to be discharged at 16 different payments (in arithmetical progression,) the first payment is to be 147. the last 100l.: What is the common difference, and the sum of the whole debt? Ans. 5l. 14s. 8d. common difference, and 9121. the whole debt. PROBLEM III. Given the first term, last term, and common difference, to find the number of terms. RULE. Divide the difference of the extremes by the common difference, and the quotient increased by I is the number of terms. EXAMPLES. 1. If the extremes be 3 and 45, and the common difference 2; what is the number of terms ? Ans. 22. 2. A man going a journey, travelled the first day five miles, the last day 45 miles, and each day increased his journey by 4 miles; how many days did he travel, and how far? Ans. 11 days, and the whole distance travelled 275 miles, GEOMETRICAL PROGRESSION, Is when any rank or series of numbers increased by common multiplier, or decreased by one common divi as 1, 2, 4, 8, 16, &c. increase by the multiplier 2; 27, 9, 5, 1, decrease by the divisor 3. PROBLEM I. The first term, the last term (or the extremes) and ratio given, to find the sum of the series. RULE. Multiply the last term by the ratio, and from the duct subtract the first term; then divide the remain by the ratio, less by 1, and the quotient will be the = of all the terms. EXAMPLES. 1. If the series be 2, 6, 18, 54, 162, 486, 1458, = the ratio 3, what is its sum total ? 1 3×1458-2 2186 the Answer. 3-1 2. The extremes of a geometrical series are 1 a 65536, and the ratio 4; what is the sum of the series F Ans. 87381. PROBLEM II. Given the first term, and the ratio, to find any other ter assigned.* CASE I. When the first term of the series and the ratio are equa *As the last term in a long series of numbers is very dicus to be found by continual multiplications, it will necessary for the readier finding it out, to have a seri of numbers in arithmetical proportion, called indice whose common difference is 1. When the first term of the series and the ratio are eque the indices must begin with the unit, and in this case, t EXAMPLES 1. The extremes are 3 and 20, and the number of ms 14, what is the common difference? -3 Extremes. Sumber of terms less 1=13)26(2 Ans. . A debt is to be discharged at 16 different paymenta arithmetical progression,) the first payment is to be the last 100l.: What is the common difference, and sum of the whole debt? ns. 5l. 14s. 8d. common difference, and 9121. the whole PROBLEM III. the first term, last term, and common difference, to RULE. vide the difference of the extremes by the common ence, and the quotient increased by I is the number rms. EXAMPLES. 1 Ans. 22. If the extremes be 3 and 45, and the common dif- ow far? 11 days, and the whole distance travelled 275 miles, |