shilling, till the last payment be £2 11s. 6d. what is the debt? 51,5+5×515-5+1=£67 125. Anf. IX 2 PROBLEM 5. The extremes and the sum of the feries given, to find the common difference. RULE-Divide the product of the sum and difference of the extremes, by the difference of twice the sum of the series, and the fum of the extremes, and the quotient will be the common difference. EXAMPLE. Let the extremes be 3 and 39, and the sum 399: what is the common difference ? Sum of the extremes=39+3= 42 Diff. of the extremes=39-3=X36 PROBLEM 6. The extremes and the fum of the feries given to find the number of terms. RULE. Twice the sum of the series, divided by the fum of the extremes, will give the number of terms. EXAMPLE. EXAMPLE. J Let the extremes be 3 and 39, and the sum of the se ries 399; what is the number of terms ? Sum of the series=399 Sum of the extr. =39+3=42)798(19 the Anf. GEOMETRICAL PROPORTION. THEOREM 1. 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 square of the mean is equal to the product of the two extremes. THEOREM 2. If four quantities, 2, 6, 4, 12, are such that the product of two of them, 2 X 12, is equal to the product of the other two, 6 X 4, then are those quan. tities 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 corresponding terms of two geometrical propections are alfo propor tional. : That is, if 2:6::4: 12and:4::5:10; then will 2 × 2:6 × 4:45:12 X 10. THEOREM * And if any quantities be proportional, their squares, abes, &c. will likewise be proportional. THEOREM 5. If four quantities, 2, 6, 4, 12, are di rectly proportional, 1. Directly, 2. Inversely, 3. Alternately, 2:4:: 6:12 4. Compoundly 2:8:: 4:16 Then, 5. Dividedly, 2:4:: 4: 8 6. Mixtly, 7. By Multiplication, 8. By Divifion, :-:: 4:12 r r し Because the product of the means in each cafe, is equal to that of the extremes, and therefore the quantities are -proportional by Theorem 2. THEOREM 6.-If three numbers, 2, 4, 8, be in continued proportion, the square of the first will be to that of the second, as the first number to the third; that is, 2 X 2:4 X 4:2:8. THEOREM 7.-In any continued geometrical proportion, (1, 3, 9, 27, 81, &c) the product of the two extremes, and that of every other two terms, equally diftant from them, are equal. THEOREM 8. The sum of any number of quantities, in continued geometrical proportion, is equal to the difference of the rectangle of the second and last terms, and the square of the first divided by the difference of the first and fecond terms. GEOMETRICAL PROGRESSION. A Geometrical Progression is when a Rank or Series, of numbers, increases, or decreases, by the continual multiplication, or divifion, of some equal number. PROBLEM I. Given one of the extremes, the ratio, and the number of the terms of a Geometrical Series, to find the other extreme. RULE. Multiply, or divide (as the cafe may require) the given extreme by such power of the ratio, whose exponent ponent* is equal to the number of terms less I, and the product, or quotient, will be the other extreme. EXAMPLES. 1. If the first term be 4, the ratio 4, and the number of terms 9; what is the last term ? 1. 2. 3. 4. + 4= 8 4. 16. 64. 256. × 256=65536=Power of the ratio, whose exponent is less by 1, than the number of terms. 1 65536-8th power of the ratio. Multiply by 4=first term. 262144 last term. Or, 4848=262144=the Anf. 2. If • As the last term, or any term near the last, is very tedious to be found by continual multiplication it will often be neceffary,in order to afcertain them, to have a feries of numbers in Arithmetical Proportion, called Indices, or Exponents, beginning either with a cypher or an unit, whose common difference is one. When the first term of the series and the ratio, are equal, the indices must begin with a unit, and, in this cafe, the product of any two terms is equal tothat term fignified by the sum of their indices. 1.2.3. 4. 5. 6. & Indices or Arithmetical feries. Thus, 4.8. 16.32.64 &c. Geometrical feries (leading terms.) Now, 6+6=12=the index of the twelfth term, and 2 64X64-4096=the twelfth term. But when the first term of the feries and the ratio are different, the indices must begin with a cypher, and the sum of the indices, made choice of, must be one less than the number of terms, given in the question; because I in the indices stands over the second term, and 2 in the indices over the third term, &c. And, in this cafe, the product of any two terms, divided by the first, is equal to that term beyond the first, fignified by the sum of their indices. 3. 4. 6, &c. Indices. 1. 3. 9. 27. 81. 243. 729, &c. Geometrical series. Thus Ο Ι. 2. Here, 6+5. 5. 11 the Index of the 12th term. 729×243-177147 the 12th term, because the first term of the feries and the ratio are different, by which mean a cypher stands over the first term, Thus, by the help of these indises, and a few of the first terms in any geometrical feries, any term, whose distance from the first term is affigned, though it were ever so remote, may be obtained without producing all the terms. 2. If the last term be 262144, the ratio, 4, and the number of terms 9; what is the first term ? 8th power of the ratio 48=65536)262144= the first 262144 Or,4 the firit term. 8 4 (term.. Again, Given the first term, and the ratio to find any other term assigned. RULE I. When the indices begin with an unit. 1. Write down a few of the leading terms of the series, and place their indices over them. 2. Add together such indices, whose sum shall make up the entire index to the term required. 3. Multiply the terms of the geometrical series, belonging to those indices, together, and the product will be the term fought 1. If the first term be 2, and the ratio 2; what is the 13th term ? 1. 2. 3. 4.5+ 5+3= 13 2.4 8. 16. 32 X 32 X 8 = 8192 Ans. Or, 2X21 8192. 2. A merchant wanting to purchase a cargo of horfes for the Westindies, a jockey told him he would take all the trouble and expense, upon himself, of collecting and purchasing 30 horfes for the voyage, if he would give him what the last horse would come to by doubling the whole number by a half penny, that is, two farthings for the first, four for the second, eight for the third, &c. to which the merchant, thinking he had made a very good bargain, readily agreed: Pray, what did the last horse come Note. If the ratio of any geometrical series be double, the difference of the greatest and leaft terms is equal to the fum of all the terms, except the greatest: If the ratio be triple the difference is double the fum of all but the greatest: If the ratio be quadruple, the difference is triple the sum of all but the greateft, &c In any geometrical series decreasing, and continued od in. finitum, balf the greatest term is equal to the sum of all the remaining terms, ad infinitum. |