ly payments in arithmetical progression, the first being $12, and the last, or fifty-second, payment $1236? Ans. 32448. 5. What debt can be discharged in one year, by weekly payments in arithmetical progression--the first being $12, and the last $1236; what is the common difference? Ans. $24. 6. A man traveling a journey, went 18 miles the first day, and increased his distance each day by 2 miles; and the last day went 48 miles. How many days did he travel, and what distance? 16 days. Ans. {528 miles. 7. A house was leased for 7 years at $400 per annum, and the rent unpaid until the end of the lease; how much was then due, simple interest, at 6 per cent. ? Ans, $3304. N. B. At the end of the 1st year, $400 was due; at the end of the second year, $424 more; at the end of the 3rd year, $448 more, &c. 8. What will be the amount of an annuity of $50, to be paid annually, but forborne 20 years; simple interest, at 6 per cent.? Ans. $1570. 9. Suppose 100 apples were placed in a right line, 2 yards apart, and a basket 2 yards from the first; how far would a boy travel to gather them up singly, and return with each separately to the basket. Ans. 20200 yards. (ART. 143.) An arithmetical series may be represented by the surface between two converging lines. Two such surfaces put together, the widest end of one against the narrowest end of the other, form a regular parallelogram. 3 5 7 9 11 Thus,. 14 14 14 14 14 The sum of the two converging spaces make one equal in width, &c. This is another method of explaining Rule 2. 10. How many acres in a piece of land 80 rods wide at one end, and 60 at the other, and 120 rods long? Ans. 52. It may be observed, that the natural numbers, 1, 2, 3, 4, 5, 6, 7, &c., is an arithmetical series, whose first term is 1, and common difference 1; and that the last term is equal to the number of terms. From this series, we may form another by adding to each term the sum of all the preceding, and we shall have 1, 3, 6, 10, 15, 21, 28, &e. These are called triangular numbers, because they may be represented by points, forming equilateral triangles, thus: Hence we perceive, that the sum of the natural numbers, to any degree, expresses the triangular number of the same degree. GEOMETRICAL PROGRESSION. (ART. 144.) A SERIES of numbers, increasing or decreasing by a common ratio, is called a Geometrical Progression. Thus, 2, 4, 8, 16, 32, 64, 128, is an increasing series, whose common ratio is 2; And 729, 243, 81, 27, 9, 3, is a decreasing series, whose common ratio is . In arithmetical progression, the terms vary by constant additions or subtractions; here, they vary by constant multiplications or divisions. The first term exists independently of the ratio. After we use the ratio once we have two terms, using it twice, we have 3 terms, &c. Wherever we stop is the last term; hence, to find the last term we have this RULE. Raise the ratio to a power one less than the number of terms, and multiply it by the first term. To investigate a rule to find the sum of any series, take the following, 3, 12, 48, 192, 768, 3072, 12288. Multiply by the ratio 4, and place the product one term to the right, 12, 48, 192, 768, 3072 12288, 49152. Subtract the upper series from the lower, and we have 49152-3. This difference comprises 3 times the original series, because we multiplied it by 4, and then subtracted the original, 4—1=3. Therefore, divide this remainder by 3, and we have the sum of the series. The remainder, 49152-3, consists of but two terms. One is the last term of the original series, multiplied by the ratio. The other term is the first term of the original series subtracted. We then divide the difference by the ratio less one, and we have the sum of the series. Hence, to find the sum of a series, we have the following RULE 2. Multiply the last term by the ratio; and from the product, subtract the first term: then divide the remainder by the ratio less one. When the extreme terms, and number of terms, are given to find the ratio, the reverse of Rute 1 is used. RULE 3. Divide one extreme by the other, and extract such a root of the quotient as corresponds to the number of terms less one. EXAMPLES. 1. The first term of a series is 4, the ratio 4, and the number of terms 9; what is the last term? Ans. 262144. 2. The first term of a series is 2, the ratio 3, and the number of terms 8; what is the last term, and what is the sum of the terms? Last term, .4374. Ans. {Sum of the terms, 6560. 3. What is the sum of ten terms of the series 1,,, &c. Ans. 174075 4. The first term of a series is 3, the last term 12288, and the number of terms 8; what is the ratio, and what are the other terms? Ans. Ratio, 4; other terms, 12, 48, &c. 5. Sold 10 yards of velvet, at 4 mills for the first yard, 20 for the second, 100 for the third, &c.; what did the piece cost? Ans. $9765,624. 6. What is the cost of a coat, with 14 buttons, at 5 mills for the first, 15 for the second, 45 for the third, &c. Ans. $11957,42. 7. What is the cost of 16 yards of cloth, at 3 cents for the first yard, 12 for the second, 48 for the third, &c.? Ans. $42949672,95. 1 32 (ART. 145.) The sum of a geometrical series is found by the extremes and the ratio, independent of the number of terms; hence, whether the number of terms be many or few, there is no variation in the rule. We may, therefore, require the sum of the series, 6, 3, 1, ¦‚ \, &c. to infinity, provided we can determine the value of the other extreme. Now, we see, the terms decrease as the series advances; and the hundredth term, for example, would be exceedingly small, the thousandth too small to be estimated, the millionth still less, and the infinite term nothing; not, as some tell us, "extremely small," or, "too little to be considered," &c.; but absolutely nothing. Remember, that it is the number of terms, not the sum of them, that is infinite. Let every decreasing series be inverted, and the first term called the last; then the first term will be 0, and the ratio greater than unity. Then, by Rule 2, work the following EXAMPLES. 1. What is the sum of the infinite series, +++ , &c.? Invert the series: is the last term, and 2 the ratio; hence, X2-0 =1, the answer. 3 2. What is the sum of the infinite series, 3, 1000, 3X10-0 &c.? 9 3. What is the value of, 25, T25, &c., to infinity? Ans. 4. What is the value of 3,,,, &c., to infinity? 2 Ans.. 5. What is the value of 1, 3 9 , &c., to infinity? 8 Ans. 4. 6. What is the value of 3, 3, 3, &c., to infinity? Ans.. 7. What is the value of ,777, &c., to infinity? This may be expressed by 7, 1 TOT, &c. Ans. 7. 8. What is the sum of ,6666, &c., to infinity? 1009 0001 Ans. 3. 9. What is the value of ,232323, &c., infinitely extended? This may be expressed by 100, 1000, 23 23 &c. Ans. 33. 23 10. What is the value of ,71333, &c., to infinity? Observe, that the geometrical series does not commence, until we pass 7. The first 3 is, the 2d is T03: ratio 10. 71 300 : 100 Ans. Sum of the series, whole value, +300 =130 107 (ART. 146.) Geometrical progression is used in finding the amount of annuities, at compound interest, when remaining a number of years unpaid. The annuity is the first term of a series. The next term is the first term, with one year's interest. This amount now becomes principal; and this again, with one year's interest, is the next term, and so on. Now to find the amount of any sum for one year, we multiply it by the amount of 1 dollar, for one year; then, of course, this multiplier is the ratio to the series, and the sum is found by the combination of Rule 1 and Rule 2, of Art. 144. But, to be less general, we give the following RULE. Raise the ratio to the power denoted by the number of years; multiply that power by the annuity; from the product subtract the annuity, and divide the remainder by the ratio less 1. |