CASE I. One of the extremes, the common difference, and the number of terms being given, to find the other extreme. 1. The first term of an increasing series is 2, the common difference 3, and the number of terms 10; what is the last term? 10 – 1= 9; (9 X 3) + 2 = 29, the last term. ANALYSIS.—As there are ten terms, we have 9 differences of 3, or 9 X3=27; the last term is therefore equal to 27 more than the first, or 27 +2= 29. 2. The first term of a decreasing series is 29, the common difference is 2, and the number of terms 10; what is the last term? 10—1= 9; 29 — (9 x 3) = 2, the last term. ANALYSIS.—As there are 9 differences of 3, or 9 X 3= 27, the last term is therefore equal to 27 less than the first, or 29 – 27= 2. RULE. To the less extreme add the product of the common difference by the number of terms less one; or, subtract this product from the greater extreme; the result in either case will be the other extreme. 3. The first term of an increasing series is 4, the common difference 5, and the number of terms 30; what is the last term? 4. The last term of a decreasing series is 300, the common difference 5, and the number of terms 40; what is the first term? 5. The first term of an increasing series is 25, the common difference 25, and the number of terms 25; what is the last term? 6. The last term of an increasing series is 192, the common difference is 1, and the number of terms is 20; what is the first term? 7. An arithmetical progression has 304 for the first term of a decreasing series, the common difference is 4, and the number of terms is 36; what is the last term ? CASE II. To find the common difference, when the extremes and the number of terms are given. 1. The extremes of a series are 2 and 10, and the number of terms is 5; what is the common difference? 10 - 2=8,84= 2, the common difference. ANALYSIS.—It has been shown in Case I, that the difference between the extremes is equal to the number of terms less one multiplied by the common difference; from this it follows that the difference of the extremes divided by the number of terms less one, or 8; 4= 2, is the common difference. RULE. Divide the differences between the extremes, by the number of terms less one. 2. The extremes of a series are 1 and 50, and the number of terms is 8; what is the common difference? 3. The first term is 9, the last term 54, and the number of terms 6; what is the common difference? 4. If $100 will amount in 10 years to $200, what is the rate per cent., simple interest? 5. The first term is 4, the last term 104, and the number of terms 41; what is the common difference? 6. A merchant discharged a debt in 6 monthly payments; the first time he paid $70, and the last time $170; by what amount did he increase his payments each time? . 10 pieces of cloth were sold at prices increasing by a common difference; the first sold at $1.25 a yard, the last sold at $2.15 a yard; what was the common difference? CASE III. To find the number of terms, when the extremes and the common difference are given. 1. The extremes are 5 and 20, and the common difference is 5; what is the number of terms? 20 – 5 = 15, 15 = 5= 3, 3+1= 4, the number of terms. ANALYSIS.—As the difference between the extremes is equal to the common difference multiplied by the number of terms less one, it follows that by dividing the difference between the extremes, 15, by the common difference, 3, and adding 1 to the quotient, we obtain 4 as the number of terms. RULE. Divide the difference between the extremes by the common difference, and add 1 to the quotient. 2. The extremes are 10 and 100, and the common difference is 10; what is the number of terms? 3. The extremes are 21 and 9, and the common difference is ; what is the number of terms? 4. If a body falls by the force of gravity 16-1, feet in the first second, and gains 32 feet each succeeding second, how many seconds has a cannon-ball been falling, if it has passed through 2091 feet during the last second of its fall? 5. In what number of years will $100 amount to $200, at 10 per cent., simple interest? 6. The first term is 200, the last term 100, and the common difference 10; what is the number of terms? CASE IV. To find the sum of the series, when the extremes and the number of terms are given. 1. The extremes are 2 and 10, and the number of terms is 5; what is the sum of the series? (10 + 2) Tax 5= 30, the sum of the series. ANALYSIS.—The com2+ 4+ 6 + 8 + 10 = 30 mon difference is found 10+ 8 + 6 + 4+ 2 = 30 to be 2. Writing the series in full and then in 12 + 12 + 12 + 12 + 12 = 60 reverse order, we take the sum of each pair of terms, and, adding these sums together, find that the amount 60, which is twice the sum of the series, is equal to 5 times (10 + 2) the sum of the extremes, and the sum of the series is =30. RULE. Multiply the sum of the extremes by the number of terms, and divide the product by two. 2. The first term is 10, the last term 60, and the number of terms 26; what is the sum of the series? 3. The extremes are 5 and 25, and the number of terms 26; what is the sum of the series? 4. The extremes are 16 L, and 2091, and the number of terms is 7; what is the sum of the series? 5. A body falls 164, feet the first second, gains 324 feet each second after the first, and falls for 13 seconds; what is its final velocity, and how far does it fall? 6. A number of telegraph poles are to be planted 75 yards apart; how far will a man travel who starts at the first one and plants ten of them, returning once from each pole to the starting point? GEOMETRICAL PROGRESSION. A Geometrical Progression is a series of numbers, each of which varies from the preceding in a constant ratio. The numbers which form the series are called the Terms. The first term and the last term are called the Extremes, and the other terms, the Means. The Ratio is the constant multiplier by which the successive terms are obtained. An Increasing Series is one in which the ratio is greater than unity. A Decreasing Series is one in which the ratio is a proper fraction. · An Infinite Series is a decreasing series in which the number of terms is infinite. The following five quantities are considered in geometrical progression: The first term; the last term; the ratio; the number of terms; and the sum of the series. CASE I. One of the extremes, the ratio, and the number of terms being given, to find the other extreme. 1. The first term of a geometrical series is 3, the ratio 2, and the number of terms 12; what is the last term? ANALYSIS.—The second term will be 3 x 2, the third term 3 x 22, and the 12th or last term 3 x 211 or 6144. |