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GEOMETRICAL PROGRESSION, OR CON
TINUAL PROPORTIONAL S. A SERIES of numbers whose successive terms increase or diminish uniformly in the same ratio, is called a PROGRESSION BY QUOTIENT, or GEOMETRICAL PROGRESSION. The numbers may also be regarded as a series of CONTINUAL PROPORTIONALS. The ratio has already been defined in the Chapter on Proportion. As in Arithmetical Progression, the first and last terms are called the extremes, the other terms the means. Thus in the ascending progression,
1, 3, 9, 27, 81, 243, the extremes are 1 and 243, and the ratio is 3. In the descending progression,
24, 12, 6, 3, , , , the extremes are 24 and is, and the ratio is .
From the nature of the series, it is evident that any four successive terms of a Geometrical Progression, constitute a proportion; as 1:3 ::9:27, 3:9:: 27 : 81, &c., in the first of the above series ; 12:6:: 3:3, 6:3:::},&c. in the second series.
One of the extremes, the ratio, and the number of terms being given, to find the other extreme.
The first term of an increasing geometrical series is 2, and the ratio 3. What is the sixth term ?
The second term will be 2 x 3; the third, 2 X 3 X 3 or 2 x 32 ; the fourth, 2 X 32 X 3, or 2 X 33, and so on, to the sixth, which is 2 x 35. If the sixth term had been given, and the first required, we should evidently have been obliged to divide by 35.
RULE. Raise the ratio to a power whose index is equal to the number of terms less one. Then for the last term multiply, and for the first term divide, the given extreme by this power of the ratio.
1 2 3 4
We have already seen in Involution, that by adding the exponents of two powers of the same number, we shall obtain the exponent of their product. Thus 37 X 34 = 3; 52 x 53 =55, &c. This princi. ple will greatly assist us in finding the powers of the ratio.
What is the 17th power of 2 ? 2, 4, 8, 16. 24 x 24 = 256 28. 28 x 28 = 256 X 256 = 65536
217. In this instance we form the powers as high as the 4th power, then multiply the 4th power by itself for the 8th power,—the 8th power by itself for the 16th power,--and the 16th power by the 1st power for the 17th power. We have, therefore, after obtaining the 4th power, only three multiplications to make, instead of thirteen, which would otherwise have been necessary.
1. The first term of a geometrical series is 3, the ratio 2, and the number of terms 9. What is the last term ?
3. What is the 11th term of the series 4096, 1024, 256, &c.?
3. What is the 7th term of a series, whose first term is 20, and ratio 1.06 ?
4. What is the amount of $20 for 7 years, at 6 per cent. compound interest?
5. What is the amount of $300 for 9 years, at 6 per cent. compound interest ?
6. If the first pane of glass in a window cost 1 mill, the second 2 mills, the third 4 mills, the 4th 8 mills, &c., what would be the price of the twelfth pane ?
7. If $100 were at simple interest for three years at 6 per cent., and the amount then placed at compound interest at 5 per cent., what would be the whole amount at the expi. ration of 12 years?
8. The first term is 8, the ratio ž, and the number of terms 8. What is the last term ?
9. The twelfth term is 59049, and the ratio ž. What is the first term ?
10. What principal will amount to $4489.643, in 12 years, at 5 per cent. compound interest ? +
11. If a farmer plants a grain of wheat, and each year plants the product of the preceding harvest, how much will
he harvest in the 15th year, the annual increase being 12 fold?
12. A farmer inquiring the price of a drove of 30 oxen, was told that he might have the whole drove for the price of the 20th ox, valuing the first at one cent, the second at 2 cents, the third at 4 cents, and so on, doubling the price of each ox for the price of the next. What would be the price per head at that rate ?
13. What is the amount of $275 for 9 years at 4 per cent, compound interest ?
14. What sum would amount to $300 in 10 years at 5 per cent. compound interest ?
15. What sum of money would amount to $1000 in 12 years at 6 per cent. compound interest ?
16. What sum would amount to $2500 in 3 years at 8 per cent. compound interest?
17. What would be the amount of $2300 in 13 years at 7 per cent, compound interest?
18. The first term of a geometrical series is 4194304, and the ratio 1. What is the fourteenth term ?
PROBLEM II. The extremes and ratio being given, to find the sum of the terms.
The first term of a series is 162, the last term is 2, and the ratio į. What is the sum of the series? The series is 162, 54, 18, 6, 2, the sum of which is 242.
54, 18, 6, 2, , is another series obtained by multiplying each term of the first series by the ratio. Subtracting the second series from the first, we have 161}, which is the difference between the first term and the last term multiplied by the ratio. It is also of the sum of the first series, since it is obtained by subtracting
of the series from the whole series. Then if we divide by , or the difference between the ratio and 1, we shall obtain the desired sum.
RULE. Multiply the last term by the ratio, and divide the differ. ence between the product and the first term by the difference between the ratio and 1.
19. The first term 9, the ratio ș, and the last term ziz are given. What is the sum of the series ?
20. What is the sum of 12 terms of the progression 2, 8, 32, &c. ? (The last term is found by Problem I.)
21. What is the sum of the series 2, 1, 1, 1, &c. to infinity ? (The last term in any infinite decreasing series is 0.)
22. Required the sum of the infinite series 7, 3, J, &c. ?
23. If I lay up $100 every year, to what will the whole amount in 10 years, at 6 per cent. compound interest ? (The 1st term is 100, the ratio 1.06, and the number of terms 10.)
24. What would be the amount of an annual saving of $300 for thirty years, at 6 per cent. compound interest ?
25. Sysla, the reputed inventor of the game of chess, is said to have asked as a reward, one grain of wheat for the first square on the chess-board, two for the second, and so on in geometrical progression. What would have been the amount of his reward, there being 64 squares on the board, and 9200 grains of wheat in a pint ? What would be the height of a cubical bin that would contain it, supposing the base to be 10 miles square ?
26. A blacksmith agreed to shoe a horse for the amount of 32 nails, at 1 mill for the first nail, 2 mills for the second, 4 for the third, and so on. What was his charge ? 27. What is the sum of the infinite series 6, 11, §, 33,
3, &c. ? 28. What is the sum of 20 terms of the series 10, 5, 21,
11, &c. ?
29. If a man commences at 21 years of age, and annually puts $500 at compound interest, how much will he be worth when he is 50 years old ?
The extremes and ratio being given, to find the number of terms.
Divide the last term by the first, and add one to the index of that power of the ratio, which is equivalent to the quotient.
30. A man made several payments in Geometrical Progression, each being twice as large as the preceding. The first payment was $4, and the last $512. What was the nurnber of payments, and what was the whole amount of the debt ?
The extremes and number of terms being given, to find the ratio.
RULE, Divide the last term by the first, and extract the root of the quotient which is indicated by the number of terms less
This and the preceding rule, are readily deduced from Problem I.
31. The first term in a Geometrical series is 1, and the eleventh term is 1024. What is the ratio ?
32. The first term is 256, and the fifth term 81. What is the ratio ? 33. The first term is 2, the last term
and the number of terms 6. What is the ratio ?
34. An estate of $200 amounted to $264.86, in seven years at compound interest. What was the rate per cent ?
35. At what rate per cent. will $1000 amount to $1689.48, in nine years at compound interest ?
36. Insert 2 mean proportionals between 1 and 6. (As there are to be 2 means, the number of terms is 4, and the extremes 1 and 6.)
37. Insert 4 mean proportionals between 3 and 96. 38. Insert 5 mean proportionals between 4 and 2916. 39. Insert 3 mean proportionals between 5 and 6480. 40. Insert 4 mean proportionals between 1 and 4. The remaining Problems in Geometrical Progression, are of little interest.
By observing the formulas on the following page, we shall perceive a striking analogy between Arithmetical and Geometrical Progression. The continual product of all the terms in a Geometrical series, is denoted by p.