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Hence, when the extremes, and the nuinber of terms are given, to find the sum of all the terms ;
Multiply the sum of the extremes by the number of terms, and the product will be the answer.
10. If the extremnes bo 3 and 273, and the number of terms 40, what is the Bm of all the terms? A. 5520.
11. How many times does a regular clock etrike in 12 hours ? A. 78. 12. A butcher bought 100 oxen, and gave for the first or $1, for the second $2, for the thira $3, and so on to the last ; how much did they come to at that ruie? A. $5050.
13. What is the sum of the first 1000 numbers, beginning with their natura: order, 1, 2, 3, &c.? A. 500500.
14. If a board, 18 feet long, bc 2 feet wide at uno end, and come tu a point ar the other. what are the square contents of the board ? A. 18 foot.
15. If a pioco of land, 60 rods in length, bo 20 rous wide at one cnd, and a. the niher terminato in an anglo or point, what number of squarc rous docs in contain A. 600.
16. A person, travelling into the country, went 3 milas the first day, and increased cvery day's travel 5 miles, tiil at last he went 58 milos in one day; how waily days did he travel ?
We found, in example 1, the difference of the extremos, divided by the number of terms, less l, gave the common difference ; consequently, if, in this example, we divide (58 – 3 =)55, the difference of the extremncs, by the common sillorence,5, the quotieri, 11, will be the number of terms, less 1; then, 1 + 1!
12, the number of cermg. A. 12.
Hence, when the extremes and common difference are given, to find the number of terms ;
Divide the difference of the extremes by the common difference, anil the quotient, incrcased by 1, will be the answer.
17. If the extremes be 3 and 45, and the common disforence 6, what is the number of terms ? A. 8.
18. A man, being askod how many children he had, rcplicd, that the youngest was 4 years old, and the eldest 32, the increase of the family having lcon 1 in overy 4 years; how many had he?' A. 8.
LXXXIX. Any rank or series of numbers, increasing by a constant multiplier, or decrcasing by a constant divisor, is called Geomctrical Progres sion.
Plys, 3, 9, 27, 81, &c., is an increasing geometrical serics ;
There are five terms in Geometrinal Progression; and, like Arithmetica. Pro Pecesion, any lhrce of them being given, the other two may be found, viz.
1. The first term.
”. The ratio ; that is, the multiplier or dirisor, by which TE- ide neer.
1. A man purchased a lock di sicep, consisting of 9 ; ani, hy agreenient WAS :o pay what the last sheep caino 10, at the rate of $4 for the first sheci, $!2 for the second, $36 for the third, and so on, trebling the prico to the last what did the flock cost lim?
Wo may perform this example by multiplication ; thus, 4 X3X3X3X3 X3X3X3X3 === $26244, Ans. But this process, you niust be sensible', would be, in many cases, a pery tedious on ; let os sco if we cannot abridge it, thoroby making it easier.
In the above process we discover that 4 is multiplied by 3 eight times, o're timo inss than the number of terms ; conscquently, the 8th power of the ratio 3, expressed thus, 38, nzultiplied by the first term, i, wil, produce the last lerin But, instead of raising 3 to the eth power in this manner, we neod only ruise it tu the 4th power, then multiply this 41.h power into itself'; for, in this way, wo slo, in fact, use the 3 eight lines, raising the 3 to the same power as befuro; thiis, 31
el; then, 81 X 81 = 1561 ; this, multiplied by 4, tho first lerin, gives $26244, tlie saine rcsuk as bsforo,' A. $321 h.
Hence, when the first term, ratio, and number of terms, are given, to find the last term;
I. Write down some of the leading powers of the ratio, with the numbers 1, 2, 3, &-c. over them, being their seocral indices
II. Add together the most convenient indices to make an inder less by 1 thun the number of terms sought.
III. Multiply together the powers, or numbers standing under those inilices ; and their product, multiplied by the first term, will be the term sought.
2. If the first term of a geometrical serics bc 4, and the ratio 3, what is tl.o Llth torin ?
1, 2, 3, 4, 5, indiccs. Nute. Tlie pupil will notice that the series
3, 9, 21, 81, 213, powers. } docs not comincnce with the first term, but with the ratio.
The indices 5 +3 +2=10, and tho powers under eachi, 243 x 27 x 9 59049 ; which, multiplied by the first torm, 4, makes 236196, the lith ierin voquired. A. 236;196.
3. The first term of a series, laviug 10 terms, is 4, and the ratio 3 ; what is the last torm? A. 78732.
4. A sain of money is to be divided among 10 persons; the first to have $10, the second $30, and so on, in threefold proportion ; what will the last tuve? A. $196830.
5. A boy purci:ased 18 oranges, on condizion that ne should pay only tho price of the last, reckoxing 1 cent for the first, 4 cents for tba second, 16'cents for the third, and in that proportion for tho whole • how much did ho pay for them? A $171798691,84.
6 What is thu last term of a series having 18 terms, 10 first of which is 3, and the ratio 3? A. 387420489.
7. A butcher mects a drover, who has 24 oren. The butcher inquires the price of them, and is answered, $60 per head ;, ho imnıcdiately, offers the drover $50 per head, and would take all. Tho drover says he will vot tako that ; but, if he will give him what tho last og would come to, at ? cents for the first, 4 cents for the second, and so on, doubling tho pricó to the last, ho might have the whole. What wíil the oxon amount to at that rate ?
A. $167772,16 8. A man was to travel to a certain place in 4 dnye, und to travel at whatover rate he pleased; the first day lie went 2 miles, the second 6 iniles, and so on to the last, in a threefold ratio ; luw far did ho travel the last day, and how fa: in all ?
In this example, wo may find the last term as bofurc, or find it by ailding each day's travel iogether, commencing with the firal, and proceeding to the last, thus: ? +6 + 18 +- 54 = 80 miles, lho whole distance travelled, and the last day's journey is 51 miles. But this mode of operation, in a long series, you must be sensible, would be very troublesome. Let us examine the natCure of the series, and try to invent sono sh ster method of arriving at the ramo result.
By examining tho scries 2, 6, 18, 54, we perceive that the last term, !51,) 0332, (the first tcrm,) = 52, is 2 times as large as the sum of the remaining werms; for 2 + 6 + 18 = 26; that is, 54 -2= 52 + 2 = 20; hence, i wo prodlure another terin, that is, multiply 51, the last term, ly the ratio 3 making 162, wo shall find the same true of this also ; for 102 – 2, (the tirst term,) 160=2=80, which we at first found to be the sum of the four remaining terms, thus : 2.+ 6 + 18+-54580. In both of theso operations it is curious to observe, that our divisor, (2,) cach tiine, 19 i less than thu ra. tio, (3.)
Hence, when the extremes and ratio are given, to find the sum of the series, we have the fol'owing easy
RULE. Multiply the last term by the ratio, from the product subtract the first terriz, and divide the remainder by the ratio, less 1; the quoticnt will be the sum of the series required.
9 If the extreincs be 5 and 6-100, and the ratio 6, what is the whole amount of the series?
6400 X 6 - 5
6--1 10. A sum of money is to be divided among 10 persons in such a manner that the first may have $10, the second $30, and so on, in thrcefold propórtion; what will iho last havo, and what will the whole have?
The pupil will recollect how he found the last term of the series by a foregoing rule; and, in all cases in which he is required to find the sum of the Beries, when the last term is not given, he must first find it by that rule, and then work for the sum of the series, by tho present rule.
A. The last, $196830 ; and the whole, $2952-10 11. A hosier sold 14 pair of stockings, the first at 4 cents, the second at le cents, and so on in goometrical progression; what did the last pair bring him, and what did the whole bring him? A. Last, $63772,92; wholo, $95439,36.
12. A man bought a horse, and, hy agrecment, was to givo a cont for the first nail, three for the second, &c.; ihere were four shocs, and in cach shoe ig!it nails; what did the horsc como lo at that ratc?
A. $9265100944259,20 13. At the marriage of a lady, one of the guests made her a present of a ha f-eagle, saying, that he would double it on the first day of cach succeed ing mouth throughout tho year, which he said would amount to something like $100 ; now, how much did his estimato differ from the truc amount:
A. $20375. 14 It' our pious ancesturs, who landed at Plymouth, A. D. 1620, !eing 101 in nouinber, had increased so as to double their number in every 20 ycars, now g.eat would hary brcn their population at the ind of the year 1440?
6 5 4 3 2
ANNUITIES AT SIMPLE INTERZST. 1 Xo. An annuity is a sum of money, payable every year, for acontain number of ycars, or forcver.
When the annuity is not paid at the time it becomes due, it is said to be in arrears.
The sun of all the annuities, euch as rents, pensions, &c., reruining an paid, with tho intercet on cach, for the time it has won duo, is called the amoult of the annuity.
Hence, to find the amount of an annuity ;-Calculate the interest on cuch unnuity, for the limc it has 7?mained unpaid, and find its amount : then the sum of all these sercral amounts will be the amount required.
1. If the annual re:t of a house, which is $290, remain unpaid, (that is, in azrears,) 8 years, what is the amount !
In this example, the cont of the last (811) year being paid wlrn due of course, there is no interest to be calculated on that yoar's rent.
The amount of $200 for 7 years $294
1936, AHS 2. If a men, having an annual pension of $60, receive ro part of it till tho expiration of 8 years, what is the amount then due? A. $5€0,80.
3. What would an annual salary of $600 arnount to, which remains unpaid (or in arrears) for 2 years? (1230For 3 years? (1908) For 4 years ? (2016) I'or 7 years! (4956) ' For 8 years? (5808) ' For 10 years ? 17620)
Ans. $24144 4. What is the present worth of an annuity of $600, to continue 4 years?
The present worth, (9 LXVII.,) is such a gum as, it'put at interesi, would e:nount to the given annuity; hence,
$600 = $1,06 = $566,037, present worth, lot year.
3d $600 – $1,24 = $183,870,
4th Ans, $2094,995, present worth required. Hence, to find the present worth of an annuity;-
Find the present worth of each year by itself, discounting from the time it becomes due, and the sum of all these present worths will be the answer.
5. What surn of ready money is equivalent to an annuity of $200, to come tinne 3 years, at 4 por cent.? A. $556,063.
6. What is the present worth of an annual salary of $80), to continue o yeurs (1469001) 3 years? (2146967) 5 years? (3407512) A. $72.
ANNUITIES AT COMPOUND INTEREST.
I XCI. The amount of an annuity, at simple and compound intercst, i tho suites, oxcepting the difference in interest.
Hence, to find the amount of an annuity at com pound interest;
Pripreed as in 1 XC., reckoning compound, instead of simple interest.
1. What will a salary of $200 amount to, which has remained unpaid for : years?
The amount of $200 for 2 years - $221,72
A. $636,72 2. If the annual rent of a house, which is $150, remain in arrcars for a years, what will be the amount due for that timc? A. $477,51.
Calculating the amount of the annuities in this manner, for a long periu of years, would bo tedious. This trouble will be prevented, by finding th amount of $1, or 1£, annuity, at coir pound interest, for a number of years as in the following
for any number of years, from 1 to 50.
1 1,0000 | 11 | 14,9716 | 21 | 39,9927 | 31 81,8016 41 165,0-107
10 13,1807 || 20 36,7855 | 30 79,0381 | 40 |154,7619:50 278,4241 Ij is evident, that the amount of $2 annuity is 2 times as much as ono of 81 and ons of $3, 3 times as much; hence,
To find the amount of an annuity, at 6 per cent. ;
Find by the Tuble the amount of $1, at the giten rate and uine, and multiply it by the given annuity, and the product will, be the amount required.
3 Whal is tho account of an annuity of $120, which has remained unfaid 15 vears:
The anivunt of $1, by the Table, we find to be $23,2759; therefore, $23,2759 . 21:20 = $2793,108, Ans.
: Whui will be the amount of an annga) salary of $400, wbich hae been in