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PROGRESSION

Conffts in two parts-ARITHMETICAL and GEOMETRICAL.

ARITHMETICAL PROGRESSION

Is when a rank of numbers increase or decrease regularly, by the continual adding or fubtracting of fome equal number: As 1, 2, 3, 4, 5, 6, are in Arithmetical Progreffion by the continual increasing or adding of one, and 11, 9, 7, 5: 3, I, by the continual decrease or fubtraction of two.

NOTE. When any even number of terms differ by Arithmetical Progreffion, the fum of the two extremes will be equal to the two middle numbers, or any two means equally distant from the extremes: As 2, 4, 6, 8, 10, 12, where 6+8, the two middle numbers are= 12+2, the two extremes, and 10+4 the two means = 14.

When the number of terms are odd, the double of the middle term will be equal to the two extremes, or of any two means equally diftant from the middle term: As 1, 2, 3, 4, 5, where the double of 3=5+1=2+4=6.

In Arithmetical Progreffion five things are to be obferved, viz. 1. The hrft term.

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Any three of which being given, the other two may be found.

The firft, fecond and third term given to find the fifth.

RULE. Multiply the fum of the two extremes by half the number of terms, or multiply half the sum of the two extremes by the whole number of terms, the product is the total of all the terms.

EXAMPLES.

1. How many ftrokes does the hammer of a clock ftrike in 12 hours ?

2.

12+1=13 then 13×6=78 Anf.

A man buys 17 yards of cloth, and gave for the first yard 25. and for last 10s. what did the 17 yards amount to?

Anf. £.5 25.

3: If 100 eggs were placed in a right line, exactly a yard afunder from one another, and the firft a yard from a basket, what length of ground does that man go who gathers up thefe 100 eggs fingly, returning with every egg to the basket to put it in? Anf. 5 miles, 1300 yards,

The firft, fecond and third terms given to find the fourth. RULE. From the second subtract the first, the remainder divided by the third less one gives the fourth.

EXAMPLES.

1. A man had 8 fons, the youngest was 4 years old, and the eldeft 32, they increase in Arithmetical Progreffion, what was the common difference of their ages ? Anf. 4.

32-4-28 then 28÷8-1-4 the common difference.

2. A man is to travel from Boston to a certain place in 12 days, and to go but 3 miles the firft day, increafing every day by an equal excels, fo that the laft day's journey may be 58 miles, what is the daily increase, and how many miles diftant is that place from Boston? Anf. 5 miles daily increase.

Therefore as 3 miles is the firft day's journey;

3+58 fecond ditto.

8+5=13 third ditto, &c.

The whole distance is 366 miles.

The firft, fecond and fourth terms given to find the third. RULE. From the second subtract the first, the remainder divide by the fourth, and to the quotient add 1, gives the third.

EXAMPLES.

I. A perfon travelling into the country, went 3 miles the first day, and increased every day by 5 miles, till at last he went 58 miles in one day, how many days did he travel? Anf. 12.

58—3—55 then 55÷5=11 and 11+1=12 the number of days.

2. A man being asked how many fons he had, faid, that the youngest was 4 years old, and the eldest 32, and that he increased one in his family every 4 years, how many had he?

Anf. 8.

The fecond, third and fourth given to find the first. RULE. Multiply the fourth by the third, made lefs by 1, the product fubtracted from the fecond gives the firft.

EXAMPLES.

I. A man in 10 days went from Bofton to a certain town in the country, every day's journey increasing the former by 4, and the last day he went was 46 miles, what was the firft? Anf. 10 miles.

4X10-136 then 46-36-10, the first day's journey.

2.

A man takes out of his pocket at 8 feveral times, fo many different numbers of fhillings, every one exceeding the former by 6; the last 46, what was the first?

Anf. 4.

The fecond, third and fifth given to find the firft.

RULE. Divide the fifth by the third, and from the quotient fubtract half the product of the fourth multiplied by the third lefs 1, gives the firft."

EXAMPLE.

A man is to receive £.360 at 12 feveral payments, each to exceed the former by £.4 and is willing to beftow the first payment on any one that can tell him what it is, what will that perfon have for his pains? Anf. £.8.

4 X 12-1

360-130 then 30

8. the firft payment.

The firft, third and fourth given to find the fecond.

RULE. Subtract the fourth from the product of the third, multiplied by the fourth, that remainder added to the first gives the

fecond.

EXAMPLE.

What is the laft number of an Arithmetical Progression, beginning at 6, and continuing by the increase of 8 to 20 places ? Anf. 158.

20X8-8=152 then 152+6=158, the laft number.

GEOMETRICAL PROGRESSION

Is the increasing or decreafing of any rank of numbers by fome common ratio, that is, by the continual multiplication or division of fome equal number: As 2, 4, 8, 16 increafe by the multiplier 2, and 16, 8, 4, 2 decrease by the divifor 2.

NOTE. When any number of terms is continued in Geometrical Progreffion, the product of the two extremes will be equal to any two means, equally diftant from the extremes: As 2, 4, 8, 16, 32, 64, where 64X24×32=8×16=128.

When the number of terms are odd, the middle term multiplied into itself will be equal to the two extremes, or any two means, equally diftant from the mean: As 2, 4, 8, 16, 32, where 2X32=4×16=8×8=64.

In Geometrical Progreffion the fame five things are to be observed, as in Arithmetical, viz.

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NOTE. As the last term in a long series of numbers is very tedious to come at, by continual multiplication; therefore, for the readier finding it out, there is a series of numbers made ufe of in Arithmetical Proportion, called indices, beginning with an unit, whose common difference is one, whatever number of indices you make use of, fet as many numbers (in fuch Geometrical Proportion as is given in the question) under them :

As

1, 2, 3, 4, 5,

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6 indices.

2, 4, 8, 16, 32, 64 numbers in Geometrical Proportion.

But if the first term in Geometrical Proportion be different from the ratio, the indices must begin with a cypher.

As 0, 1, 2, 3, 4, 5, 6 indices,

1, 2, 4, 8, 16, 32, 64 numbers in Geometrical Proportion.

When the indices begin with a cypher, the fum of the indices made choice of must be always one lefs than the number of terms gi ven in the question, for 1 in the indices is over the second term, and 2 over the third, &c.

Add any two of the indices together, and that fum will the product of their refpective terms.

As in the first table of indices 2+5= 7

Geometrical Proportion

Then in the fecond

4X32=128

2+4= 6

4X16 64

agree witla

In any Geometrical Progreffion proceeding from unity the ratio be ing known, to find any remote term, without producing all the intermediate terms.

RULE. Find what figures of the indices added together would give the exponent of the term wanted, then multiply the numbers ftanding under fuch exponent into each other, and it will give the term required.

NOTE. When the exponent 1 ftands over the second term, the number of exponents must be i less than the number of terms.

EXAMPLES.

1. A man agrees for peaches, to pay only the price of the laft, reckoning a farthing for the firft, a half-penny for the fecond, &c. doubling the price to the laft, what muft he give for them?

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2. A country gentleman going to a fair to buy some oxen, meets with a person who had 23, he demanding the price of them, was anfwered, £.16 apiece; the gentleman bids him £.15 apiece, and he would buy all; the other tells him it would not be taken, but if he would give what the last ox would come to, at a farthing for the firft, and doubling it to the laft, he fhould have all. What was the price of the oxen ? Anf. £.4369 15. 4d.

In any Geometrical Progreffion, not proceeding from unity, the ratio being given, to find any remote term, without producing all the intermediate terms.

RULE. Proceed as in the laft, only obferve that every produ must be divided by the firft term.

1.

EXAMPLES.

A fum of money is to be divided among eight perfons, the first to have £.20, the fecond £.60, and fo on in triple proportion, what will the last have ?

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3+3+17 one less than the number of terms.

12. A gentleman dying, left 9 fons, to whom and to his execu tors, he bequeathed his eftate in manner following: To his executor £.50; his youngest fon was to have as much more as the executor, and each fon to exceed the next younger by as much more; what was the eldest fon's portion? Anf. £.25600.

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