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GEOMETRICAL PROGRESSION, IS when any rauk or series of nunibers increase by one common multiplier, or decrease by one cominon 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.
EXAMPLES. 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 gevmetrical series are 1 and 65536, and the ratio 4; what is the sum of the series ?
Ans. 87381. 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 scries and the ratio are equal, the indices must begin with the unit, and in this case, the product of any two terms is byual to that term, signified by the sum of their indiccs :
1. Write down a few of the leading terms of the sene and place their indices over them, beginning the indicei with a unit or l.
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 tern suught.
Í. If the first be 2, and the ratio 2; what is the 13t). term ? 1, 2, 3, 4, 5, indices.
Then 5+5+3=13. 2, 4, 8, 16, 32, leading terms. 32 x 32 x8=8192 Ans.
2. A draper sold 20 yards of superfine cloth, the first yard for 3d., 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 3480784401d. Then 3+3486784401-3
5230176600d. the sum of all
the terms (by Prob. I.) equal to £21792402, 10s.
3. A rich miser thought 20 guineas a price too much for 12 fine horses, but agreed to give 4 cts. for the first, 16 cts. 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 $223696, 20 cts., and the average price was $18641, 35 cts. per head.
1 1 2 3 4 5, &c. indices or arithmetical series hus, 1 2 4 8 16 32, &c. geometrical series. No. 3+2 = 5 = the index of the fifth term, and wow, 4x8 = 32 = the fifth term.
CASE 1. 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 cipher: Thus, 0, 1, 2, 3, &c
2. Add together the most convenient indices io make an index less by 1 than the number expressing the place of the terins sought.
3. Multiply the terms of the geometrica. series together belonging to those indices, and make the product a dividend
4. Raise the 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 2, what is the 7th term ?
0, 1, 2, 3, Indices,
3+2+1=6, the index of the 7th term. 108 x 36 x 12=46656
- =2916 the 7th term required.
- 16 Here the number of terms multiplied are three; theresure the first term raised to a power less than three, is the 20 power or square of the 16 the divisor.
* When the first term of the series and the ratio are different, the in dices must begin with a cipher, and the sum of the indices made choice of biust be one less than the number of terms given in the question : because 1 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 indiccs.
e , 1, 2, 3, 4, &c. Indices.
10, 11, 3, 9, 27, 81, &c. Geometrical series. Here 4+3=7 the index of the 8thterm.
81 X 27=2197 the 8th term, or the 7th heyond the 1sl.
5. A Goldsmith sold 1 lb. of gold, at 2 cts. for the firsi 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, 10 cts. 6. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, or (2 d) the second and so on, each month in a tenfold proportion ?
Ans. £115740740 14s. 9d. 3 qrs. 7. A thrasher worked 20 days for a farmer, and received for the first days work four barley-corns, for the second 12 barley corns, for the third 36 barley corns, and so on, in triple proportion geometrically. I demand what the 20 day's labour came to supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushel ? Ans. £1773 7s. 6d. rejecting remainders
8. A man bought a horse, and by agreement, was to give a farthing for the first nail, two for the second, fivui for the third, &c. There were four shoes, and eight nails in each shoe ; what did the liorse come to at that rate ?
Ans. £4473924 5s. 3;d 9. Suppose a certain body, put in motion, should move the length of 1 barley-corn the first second rf time, a ne inch the second, and thiee inches the third ocond 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.
Ans. 95:3199685623 yds. 1 ft. 1 in. 16. which is no less than five hundred und 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 perforin ile 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 rum in the question : : so is the supposed number : to the true one required.
The inethod of proof is by substituting the answer in the ques tion.
1. A schoolmaster being asked how many scholars ho tiad, said, if I had as many more as I now have, half as many, one-third, and one fourth as many, I should thel bave 148 ; How many scholars had he ? Suppose he had 12 As 37 : 148 : : 12 : 48 Ans. as many = 12
48 as many = 6
24 as many = 4
16 as many = 3
12 Result, 37
Proof, 143 2. What number is that which being increased by 4, t. and of itself, the sum will be 125?
Ans. 67. 3. Divide 9:3 dollars between A, B and C, so thai 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 $31, B's $15., and C's 8464. 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 re. ceive 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 to him at first ?
Ans. £425. 6. A vessel has 3 cocks, A, B and C; A can fill it ini 1 bour, B in 2 hours, and C in 4 hours ; in what time will they all fill it together ?
Ans 34 min. 174 sec.