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Ex. 2. The first and last terms of a series are 3 and 111, and the number of terms 37 : what is the sum?
Ex. 3. How many strokes do the clocks of Venice strike in 24 hours, where they strike from 1 to 24?
Ex. 4. The first and last terms of a series are 1 and 1000, and the number of terms 100: required the sum.
Ex. 5. If 100 stones are placed in a right line, exactly a yard asunder, and the first one yard from a basket, what length of ground will a man go over, who gathers them up, one by one, returning with each to the basket?
Ex. 6. What must a man give for 54 timber trees, for which he pays 5 shillings for the first, and 201. for the last, and the prices of the others being in arithmetical progression ?
Ex. 7. A butcher buys a drove of oxen, consisting of 32; for the first he pays 158., and for the others he is to pay in arithmetical progression, so that for the last he is to pay 381.: what will they all come to?
Ex. 8. A horse-dealer sends to a fair 63 horses, of various kinds and worth, which he is willing to dispose of according to the principles of arithmetical progression, demanding 31. only for the fust, provided he had 531. for the last: how much did he receive for the whole, and what was the average value of cach horse?
II. The first and last terms, a and z, and number of terms
being given, to find the common difference, d. RuLÈ. The difference of the extreme terms divided by the number of terms, less 1, will be the common diference sought:
8 a 200
Ex. 1. What is the common difference of an arithmetical progression, whose extremes are 8 and 200, and the number of terms 17?
16 Ex. 2. When the extremes of an arithmetical progression are 6 and 37, and the number of terms 18, what is the common difference ?
Ex. 3. A gentleman gives at Christmas, among his 25 poor neighbours, a sum of money in arithmetical progression : to the least needy he gires 5 shillings, and to the poorest, with a very large family, he gives five guineas: what was the common difference?
* This mark wn,
is put for the difference of any two numbers, which ever of them is largest ; thus, suppose a be 10, and ž 4, then a w z = 6, or it may be, a is 4, and z 15; then a w z = 11.
Ex. 4. A traveller is out on his journey a month, of which he travels 25 days; on the first he rides 7 miles, and on the last, having little to do, he comes 43 miles : how much was the daily increase of his travelling, and how many miles did he ride in the whole ?
III. The extreme terms a and z, and common difference d
being given, to find the number of terms n. Rule. Divide the difference of the extremes by the common difference, and the quotient increased by unity is the number sought: or
+1R. Ex. l. When the extremes are 4 and 106, and the common difference is 3, what is the number of terms ?
+ 1 = +1=34 +135 = n. 3
3 Ex. 2. If the least term be 6, the greatest 216, and the common difference 5, what is the number of terms ?
Ex. 3. What debt can be paid, and in what time, supposing I agree to lay by 3s. the first week, Is. the next, 11s. the third, and so on in arithmetical progression, till the last saving be four guineas ?
Ex. 4. I set out for Hastings, which is 69 miles from this place, and I walk the first day 4 miles, the second 7, increasing every day by 3 miles, and on the last 19 miles : how many days will the journey take?
In addition to the above, the learner may commit'to memory the following facts on the subject :
1. If three numbers are in arithmetical progression, the sum of the extremes is equal to double the mean term; as 6, 9, 12, where 6 + 12 = 2 XI = 18.
2. If four numbers be in arithmetical progression, the sum of the two extremes is equal to the sum of the means ; as 5, 8, 11, 14, where 5 + 14 = 8 + 11 = 19.
3. When the number of terms is odd, the double of the middle term will be equal to the sum of the extremes ; or of any other two means equally distant from the middle term; as 3, 8, 13, 18, 23, 25, 33, where 3*+ 33 = 2 X 18
13 + 23 = 8 + 28.
A GEOMETRICAL PROGRESSION is a series of numbers, the terms of which gradually increase or decrease by ihe constant multiplication or division of some particular number; as 1, 3, 9, 27, 81, 243, &c., or 64, 32, 16, 8, 4, 2, 1, 1, &c.
In the first case, the series is increasing by the constant multiplication of 3; in the second, it is a decreasing series by the constant division of 2. It is evident that both series may be carried on for ever.
The number by which the series is constantly increased or diminished is called the ratio.
The first term is called
The sum of all the terms is called
ratio r, to find the sum s. RULE. Multiply the last term by the ratio, and from the product subtract the first term, and the remainder divided by the ratio, less one, will give the sum of the series; or EXT-a
Ex. 1. The first term of a series in geometrical progression is 5, the last term is 3645, and the ratio is 3 : what is the sum? 3645 X 3 - 5 1.0935
10930 Here s =
- 5465. 3
2 For the terms are 5, 15, 45, 135, 435, 1215, and 3645; which, being added together, make 5465.
Ex. 2. The first and last terms of a geometrical series are 4 and 32941791 and the common ratio is 7 : what is the sum ?
Ex. 3. The first and last terms of a geometrical progression are 4 and 262144, and the ratio 4: what is the sum?
II. Given the first term a, the number of terms n, and the
ratio r, to find the last term z. The last term may be obtained by continual multiplication ; but as that, in a long series, is a tedious process, we shall give the following rule:)
1. When the first or least term is equal to ratio. RULE. Write down some of the leading terms of the geometrical series, orer which place the arithmeticul series 1, 2, 3, 4, &c., as indices ;* And what figures of these indices added together will give the index of the term wanted in the geometrical series; then multiply the numbers, standing under such indices, into each other, and their product will be the term sought.
Ex. 1. What is the last term of a geometrical series having 13 terms, of which the first is 2, and the ratio 2 ? Here the series, with their indices, will stand thus :
2', 4*, s°, 16+, 32°, 64°, &c. The number of terms being 13, the index to the last term will be 13 equal to the indices + 5+ 6, which figures standing over 4, 32, and 64, shew that these last are to be multiplied together, and the product is the term sought ; thus 4 X 32 X 64 = 8192.
Ex. 2. What is the last term of the series having 9 terms, of which the first is 3, and the ratio 3 ?
Ex. 3. What did the last of 12 oxen cost, the first of which was sold for 3s., the second for gs., and so on?
2. When the first term a, of the series, is not equal to the
RULE. Write down the leading terms of the series, and place their indices over them, beginning with a cypher, add together the most convenient indices to make an index less one than the number expressing the place of the term sought; then multiply the numbers standing under such
* When the natural numbers 1, 2, 3, 4, 5, &c., are set over a geometrical series, they are called indices, or exponents, and they shew the distance of any term from unity, or from the first term : thus, in the series 2', 4°, 8°, 16*, 64", 128o, &c., 1, 2, 3, &c. are the indices, and shew the distance of any term of the series from the first term; the index 5, for instance, shews that 64 is the fifth term in the series.
indices, into each other, dividing the product of every two by the first term in the geometrical series; the lust quotient iš the term. required.
Ex. 1. What is the last term of the series, whose first term is 4, ratio 3, and number of terms 15 ?
4", 12", 36', 1089, 324*, 972", 29166, &c. The number of terms being 15, the index sought must be 14 equal to 6 + 5 + 3, under which stand the terms 2916, 972, and 108, then 2916 X 972
708588 X 108 708588, and
19131876 = x=lagt 4 term,
Ex, 2. The first term of a geometrical series is 2, the number of terms 12, and the ratio 5, required the last term ?
Ex. 3. The first term of a geometrical series is 1, the ratio 2, and the number of terms 25, what is the last term, and also the sum of all the terms ?
Ex. 4. The first term of a series is 5, the ratio 3, and the number of terms 16, what is the last term, and the sum of the terms ?
Ex. 5. A hosier sold 12 pair of stockings, the first pair at 3d., the second gd., and so on in geometrical progression; for what did he sell the last pair, and how much had he for the whole?
Ex. 6. What would a horse fetch, supposing it was sold on condition of receiving for it one farthing for the first nail in his shoes, a halfpenny for the second, one penny for the third, and so on, doubling the price of every nail to 32, the number in his four shoes ?
Ex. 7. A husbandman agreed to serve his master during hay-time and harvest, or five-and-forty clear days, provided he would give him a barley-corn only for the first day's work, 3 for the second, for the third, and so on in geometrical proportion; what would he have to receive in money for his labours, supposing there were half a million of grains in a bushel, and each bushel was worth 4s. ?
The following facts may be committed to memory : 1. If three numbers are in geometrical progression, the product of the extremes is equal to the square of the means; as 3, 9, 27, here 3 x 27 = 9 X 9 = 81.
2. If four numbers are in geometrical progression, the product of the extremes is equal to the product of the means ; as 2, 4, 8, 16; here 2 x 76 = 4 *8 = 32.
3. If the series contain an odd number of terms, the square of the middle term is equal to the product of the adjoining extremes, or of any two terms equally distant from thein; as 3, 9, 27, 81, 243; here 272 +3 X 243 9 X 81.