ponent* is equal to the number of terms lefs 1, and the product, or quotient, will be the other extreme.

1. If the first term be 4, the ratio 4, and the number of terms 9; what is the laft term?

I. 2. 3. 4. + 4=

8

4. 16. 64. 256. x 256-65536=Power of the ratio, whofe exponent is lefs by 1, than the number of terms. 65536 8th power of the ratio.

Multiply by 4 first term.

262144 laft term.

Or, 4X4262144 the Anf.

2. If

* As the last term, or any term near the laft, is very tedious to be found by continual multiplication, it will often be necessary, in order to afcertain them, to have a feries of numbers in Arithmetical Proportion, called Indices, or Exponents, beginning either with a cypher or an unit, whofe common difference is one.

When the first term of the feries and the ratio, are equal, the indires must begin with an unit, and, in this cafe, the product of any two terms is equal to that term fignified by the fum of their indices. Thus SI. 2. 3. 4. 5. 6. &c. Indices or Arithmetical feries.

22. 4. 8. 16. 32. 64. &c. Geometrical feries (leading terms.) Now. 6+6 12 the index of the twelfth term, and

=

64x64-4096 the twelfth term.

But when the first term of the feries and the ratio are different, the indices must begin with a cypher, and the fum of the indices, made choice of, must be one lefs than the number of terms, given in the question; becaufe 1 in the indices ftands over the fecon term, and 2 in the indices over the third term, &c. And, in this cafe, the product of any two terms, divided by the firft, is equal to that term beyond the firft, fignified by the fum of their indices. Thus' Sc. I. 2. 3. 4. 5. 6, &c. Indices.

1. 3. 9. 27. 81. 243. 729, &c Geometrical feries. Here, 65

11 the Index of the rth term. 729X243=177147 the 12th term, because the first term of the feries and the ratio are different, by which mean a cypher ftands over the firft term,

Thus, by the help of these indices, and a few of the first terms in any geometrical feries, any term, whofe difiance from the first term is affigned, though it were ever fo remote, may be obtained without producing all the terms.

Note.

2. If the last term be 262144, the ratio, 4, and the number of terms 9; what is the first term?

8th power of the ratio 48=65536)262144 the firft

Or,

262144

4 the firft term.

(term.

Again, Given the firft term, and the ratio to find any other term affigned.

RULE 1. When the indices begin with an unit.

1. Write down a few of the leading terms of the feries, and place their indices over them.

2. Add together fuch indices, whofe fum fhall make up the entire index to the term required.

3. Multiply the terms of the geometrical feries, belonging to those indices, together, and the product will be the term fought.

1. If the firft term be 2, and the ratio 2; what is the 13th term?

1. 2. 3. 4. 5 + 5 + 3

13

2. 4. 8. 16. 32 X 32 X 8 = 8192 Anf.

Or, 2X2128192.

2. A merchant wanting to purchase a cargo of horses for the Weftindies, a jockey told him he would take all the trouble and expenfe, upon himfelf, of collecting and purchafing 30 horfes for the voyage, if he would give him what the laft horfe would come to by doubling the whole number by a half penny, that is, two farthings for the firft, four for the fecond, eight for the third, &c. to which the merchant, thinking he had made a very good bargain, readily agreed: Pray, what did the laft horfe

come.

Note. If the ratio of any geometrical feries be double, the difference of the greatest and leaft terms is equal to the fum of all the terms, except the greatest: If the ratio be triple, the difference is double the fum of all but the greatest: If the ratio be qua truple, the difference is triple the fum of all but the greateft, &c.

In any geometrical ferics deereafing, and continued ad infinitum, half the greatest term is equal to the fum of all the remaining terms, ad infinum.

come to; and, what did the horses, one with another, coft the merchant?

1. 2. 3. 4. 5. 6+ 6= 12th.

[30th, or laft term.

12+12+6=

2. 4. 8. 16. 32. 64×64=4096&4096 × 4096 × 64— 1073741824 yrs. =£1118481 1s. 4d., and their average price was £37232 145. od. a piece.

RULE 2. -When the indices begin with a cypher.

1. Write down a few of the leading terms of the feries, as before, and place their indices over them.

2. Add together the moft convenient indices to make an Index, lefs by I than the number expreffing the place of the term fought.

3. Multiply the terms of the geometrical feries, together, belonging to thofe indices, and make the product a dividend.

4. Raife the firft term to a power, whofe Index is one lefs than the number of terms multiplied, and make the refult a divifor, by which, divide the dividend, and the quotient will be that term beyond the first, fignified by the fum of thofe indices, or the term fought.

3. If the first term be 5, and the ratio 3, What is the 7th term.

6 Ind. to the term be. (yond it, or 7th.

5. 15. 45. 135 × 45 × 1591125 Dividend.

The number of terms, multiplied, is 3 (viz. 135 X45 ×15) and 3-12, is the power to which the term 5 is to be raised; but the 2d power of 5 is 5×525, and therefore 91125-25=3645 the 7th term required.

PROB. 2. Given the firft term, the ratio, and number of terms, to find the fum of the feries.

RULE.-Raife the ratio to a power, whofe Index fhall be equal to the number of terms, from which fubtract; divide the remainder by the ratio, lefs

I, and the quotient, multiplied by the firft term, will give the fum of the fertes.

EXAMPLES.

1. If the first term be 5, the ratio 3, and the number of terms 7; What is the fum of the feries?

Ratio 3×3×3×3×3×3×3=21877th power I (of the ratio.

Subtract

Divide by the ratio less 1=3—1=2)2186

Or, 35=5465 Answer.

2. A fhopkeeper fold 13 yards of cloth, on the following terms, viz. 2d. for the first yard, 4d. for the fecond, 8d. for the third &c. I demand the price of the cloth.

2

2

I

I *× 2=16382d.=£68 5s. 2d. Anfwer.

3. A gentleman, whofe daughter was married on a new year's day, gave her a guinea, promifing to triple it on the first day of each month in the year; Pray what did her portion amount to?

3

3

I X 1265120 guineas, Anfwer.

I

4. What debt can be difcharged in a year, by paying I fhilling the first month, 10s. the fecond, and fo on, each month in a tenfold proportion?

-10 IXI=III11111111 fhillings, £55555555555 11s. Anfwer.

5. A man threshed wheat 9 days for a farmer, and agreed to receive but 8 wheat corns for the first day's work, 64 for the fecond, and fo on in an eight fold proportion; I demand what his 9 days' labour amounted to, rating the wheat at 5s. per bushel ?*

Anf.8153391688 corns. Amount = £78 os. (5 d.

6. An ignorant fop wanting to purchase an elegant houfe a facetious gentleman told him he had one which he would fell him on these moderate terms, viz. that he fhould give him a penny for the first door, 2d. for the fecond, 4d. for the third, and fo on, doubling at every door, which were 36 in all: It is a bargain, cried the fimpleton, and here is a guinea to bind it; Pray what would the houfe have coft him?

X1=68719476735d.=£286331153 is. 3d.

7. A young fellow, well fkilled in numbers, agreed with a rich farmer to ferve him 10 years, without any other reward, but the produce of one wheat corn for the firft year, and that produce to be fowed from year to year, till the end of the time, allowing the increafe but in a tenfold proportion; What is the fum of the whole produce ? and, what will it amount to, at 55. per bufhel?

Amount =

(£5651 8s. 0, d.

8. Suppofe one farthing had been put out at 6 per cent. per annum, compound intereft, at the commence

ment

*

Note, 7680 wheat or bailey corns are fuppofed to make a pint + Any fum, at £6 per cent. per annum, compound intereft wil double in eleven years and three hundred and twenty five days, or 11,889 years, or 11,89 is near enough; then, if you divide 1784 by ,! 1,89, it will give the number of terms in this cafe equal to 150;— The ratio will be 2, and the first tegm 1.