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GEOMETRICAL PROGRESSION, Is when any rank or series of numbers increased by one common multiplier, or decreased by one common divisor as 1, 2, 4, 8, 16, &c. increase by the multiplier 2; and 27, 9, 3, 1, decrease by the divisor 3.
PROBLEM I. The first term, the last term (or the extremes) and the ratio given, to find the sum of the series.
RULE. Multiply the last term by the ratio, and from the product subtract the first term; then divide the remainder by the ratio, less by 1, and the quotient will be the sum of all the terms.
1. If the series be 2, 6, 18, 54, 162, 486, 1458, and the ratio 3, what is its sum total ?
-2186 the Answer. ;
3-1 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4; what is the sum of the series ?
Ans. 87881. PROBLEM II. Given the first term, and the ratio, to find any other term
CASE I. When the first term of the series and the ratio are equal.
* As the last term in a long series of numbers is very tedious to be found by continual multiplications, it will be necessary for the readier finding it out, to have a series of numbers in arithmetical proportion, called indices, whose common difference is 1.
When the first term of the series and the ratio are equal, the indices must begin with the unit, and in this case, the 1. Write down a few of the leading terms of the series, and place their indices over them, beginning the indices with an unit or 1.
2. Add together such indices, whose sum shall make up the entire index to the sum required.
3. Multiply the terms of the geometrical series belonging to those indices together, and the product will be the term sought.
1. If the first be 2, and the ratio %; what is the 13th term. 1, 2, 3, 4, 5, indices. Then 5+5+3=13 2, 4, 8, 16, 32, leading terms. 32x32x8=8192 Ans.
2. A draper sold 20 yards of superfine cloth, the first yard for Sd. the second for 9d. the third for 27d. &c. in triple proportion geometrical ; what did the cloth come to at that rate ?
The 20th, or last term is 3486784401d. Then 3+3486784401.-3
=5230176600d. the sum of all
3-1 the terms (by Prob. I.) equal to £21792402 10s. Ans.
3. A rich miser thought 20 guineas a price too much for 12 fine horses, but agreed to give 4 cents for the first, 16 cents for the second, and 64 cents for the third horse, and so on in quadruple or fourfold proportion to the last: what did they come to at that rate, and how much did they cost per head, one with another ?
Ans. The 12 horses came to 8223696, 20cts, and the average price was 818641, 35cts. per head.
product of any two terms is equal to that term, signified by the sum of their indices.
Ş 1 2 3 4 5 &c. Indices or arithmetical series. 2 2 4 8 16 32 &c. geometrical series.
5 = 02. 5+2
the index of the fifth term, and 4 X8 32 = the fifti term.
CASE II, When the first term of the series and the ratio are diffe.
rent, that is, when the first term is either greater or less than the ratio. *
1. Write down a few of the leading terms of the series, and begin the indices with a cypher: Thus, 0, 1, 2, 3, &c.
2. Add together the most convenient indices to make an index less by 1 than the number expressing the place of the term sought.
3. Multiply the terms of the geometrical series together belonging to those indices, and make the product a dividend.
4. Raise tłe first term to a power whose index is one less than the number of the terms multiplied, and make the result a divisor.
5. Divide, and the quotient is the term sought.
4. If the first of a geometrical series be 4, and the ratio 3, what is the 7tn term ? 0, 1, 2, 3, Indices. 4, 12, 36, 108, leading terms.
3+2+1=6, the index of the 7th term. 108X36X12=46656
-=2916 the 7 th term required
16 Here the number of terms multiplied are three; therefore the first terin raised to a power less than three, is the 2d power or square of 4=16 the divisor.
*When the first term of the series 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 case, the product of any two terms, divided by the first, is equal to that term beyond the first, signified by the sum of their indices. Thus,
0, 1, 2, 3, 4, c. Indices.
1, 3, 9, 27, 81, 8c. Geometrical series. Here 4+3=7 the index of the 8th term.
81 X27=2187 the 8th term, or the 7th beyond the 1st.
5. A Goldsmith sold 1 lb. of gold, at 2 cents for the first ounce, 8 cents for the second, 32 cents for the third, &c. in a quadruple proportion geometrically; what did the whole come to ?
Ans. $111848, 10cts. 6. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, (or 23d.). the second, and so on, each month in a tenfold proportion ?
Ans. £115740740 14s. 9d. 3.jrs, 7. A thresher worked 20 days for a farmer, and receive ed for the first day's work four barley-corns, for the second 12 barley-corns, for the third 36 barley-corns, and so on in triple proportion geometrical. I demand what the 20 days' labor came to, supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 23. 6d. per bushel ? Ans. £1773 7s. 6d. rejecting remuinders.
8. A man bought a horse, and by agreement was to give a farthing for the first nail, two for the second, four for the third, &c. There were four shoes, and eight nails in each shoe; what did the horse come to at that rate ?
Ans. £4473924 5s. 3id. 9. Suppose a certain body, putin motion, should move the length of one barley-corn the first second of time, one inch the second, and three inches the third second of time, and so continue to increase its motion in triple proportion geometrical; how many yards would the said body move in the term of half a minute ?
Ars. 953199685623 yds. ift. lin. 1b.c. which is no less than five hundred and forty-one millions of miles.
POSITION. POSITION is a rule which, by false or supposed num. bers, taken at pleasure, discovers the true ones required. It is divided into two parts, Single or Double.
SINGLE POSITION, Is when one number is required, the properties of which are given in the question.
RULE. 1. Take any number and perform the same operation with it, as is described to be performed in the question.
2. Then say; as the result of the operation : is to the given sum in the question : : so is the supposed number : to the true one required.
The method of proof is by substituting the answer in the question.
1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I'now have, half as many, one-third and one-fourth as many, I should then have 148; How many scholars had he?
Suppose he had 12 As 37 : 148 : : 12 : 48 Ans.
16 $ as many
12 Result, 37
Proof, 148 2. What number is that which being increased by 1, j; and I of itself, the sum will be 125 ?
Ans. 60. 3. Divide 93 dollars between A, B and C, so that B's share
may be half as much as A's, and C's share three times as much as B’s.
Ans. A's share 831, B's $154, and C's $461. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 3 times as much as A, and C took up as much as A and B both; what share of the gain had each ?
Ans. A $40, B $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 6l. per cent. per annum, simple interest, and at the end of twelve years received 7311. principal and interest together; what was the sum delivered him at first ?
Ans. £425. 6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, B in 2 hours and C in 4 hours ; in what time will they all fill it together? Ans. 34min. 174 sec.