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ARITHMETICAL PROGRESSION Is when a rank of numbers increase or decrease regularly, by the continual adding or fubtracting of some equal number : As 1, 2, 3, 4, 5, 6, are in Arithmetical Progression by the continual increasing or adding of one, and 11,9,7, 5, 3, 1, by the continual decrease or subtraction of two.
Nore. When any even number of terms differ by Arithmetical Progression, the sum 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 estremes, and = 10+4 the two means = 14.
Woen the number of terms are odd, the double of the middle terin will be equal to the two extremes, or of any two means equally distant from the middle term : As 1, 2, 3, 4, 5, where the double of 3=5+1=2+4=6. In Arithmetical Progression five things are to be observed, viz.
The first term.
The last term.
The numb'r of terms.
The surn of all the terms.
The first, fecond and third term given to find the fifth. RULE. Multiply the sum 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.
strokes does the hammer of a clock Atrike in 12 hours ?
12+1=13 then 13 X6=78 Ans. 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 ?
Ans. £.5 25. 3. If 100 eggs were placed in a right line, exactly a yard asunder' from one another, and the first a yard from a basket, what length of ground does that man go who gathers up these 100 eggs singly, returning with every egg to the baiket to put it in ?
Anl. 5 miles, 1300 yards,
The firji, second 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.
A man had 8 fons, the youngest was 4 years old, and the eld. eft 32, they increase in Arithmetical Progreslion, what was the common difference of their
A man is to travel from Boston to a certain place in 12 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 58 miles, what is the daily increase, and how many miles distant is that place from Boston?
Anf. 5 miles daily increase.
3+5= 8 second ditto.
The first, second 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. 1. A person 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 ? 58—3=55 then 55+5=11 and 11 +1=12 the number of days.
A man being asked how many sons he had, said, 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?
The second, third and fourth given to find the first. RULE. Multiply the fourth by the third, made less by 1, the produa subtracted from the second gives the first,
EXAMPLES. A man in 10 days went from Boston 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 first ? Anf. 10 miles,
4X10-1=36 then 46-36=10, the first day's journey,
2. A man takes out of his pocket at 8 several times, so many different numbers of shillings, every one exceeding the former by 6 ; the last 46, what was the first ?
The second, third and fifth given to find the first. RULE. Divide the fifth by the third, and from the quotient fub. tract half the product of the fourth multiplied by the third less 1, gives the firlt.
EXAMPLE. A man is to receive £ -360 at 12 several payments, each to exceed the former by £ .4 and is willing to bestow the first payment on any one that can tell him what it is, what will that person have for his pains ?
360-19=30 then 30
4 X 12-1
=8. the first payment.
The first, third and fourth given to find the second. 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 last number of an Arithmetical Progression, be. ginning at 6, and continuing by the increase of 8 to 20 places ?
20X8--8=152 then 152+6=158, the last number.
GEOMETRICAL PROGRESSION Is the increasing or decreasing of any rank of numbers by some common ratio, that is, by the continual multiplication or division of some equal number: As 2, 4, 8, 16 increase by the multiplier 2, and 16, 8, 4, 2 decrease by the divisor 2.
Note. When any number of terms is continued in Geometrical Progression, the product of the two extremes will be equal to any two means, equally distant from the extremes : As 2, 4, 8, 16, 32, 64, where 64 X2=4X32=8 X 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 distant from the mean : As 2, 4, 8, 16, 32, where 2X32=4x16=8X8=64.
In Geometrical Progression the same five things are to be observed, 23 in Arithmetical, viz.
The first term,
Nork. 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 use of in Arithme. tical Proportion, called indices, beginning with an unit, whose common difference is one, whatever number of indices you make use of, set as many numbers in such Geometrical Proportion as is given in the question) under them : As
1, 2, 3, 4, 5, 6 indices.
But if the first term in Geometrical Proportion be different fron 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 sum of the indices made choice of must be always one less than the number of terms given 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 sum will agree with the product of their respective terms.
As in the first table of indices 24 s=
3+ 4= 6
In any Geometrical Progression proceeding from unity the ratio being 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 standing under such exponent into each other, and it will give the term required.
Note. When the exponent i stands 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 last, reckoning a farthing for the first, a half-penny for the second, &c. doubling the price to the last, what must he give for them?
16=4 0, 1, 2, 3, 4, exponents.
16=4 1, 2, 4, 8, 16, number of terms.
4+4+3=11, number of terms lefs 1.
4)2048=11 numb, farth.
£.2 2 8 Answer.
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 anSwered, £ .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 first, and doubling it to the last, he should have all. What was the price of the oxen?
Anf. £ -4369 Is. 4d.
In any Geometrical Progression, not proceeding from unity, the ratio being given, to find any remote term, without producing all the intermediate terms.
Rule. Proceed as in the last, only observe that every produa must be divided by the first term.
EXAMPLES. 1. A sum of money is to be divided among eight persons, the first to have £.20, the second £.60, and so on in triple proportion, what will the last have ?
540 X 540
14580 X 60
-43740 20. 6o. 180. 540. 20
Anf. £.43740. 3+3+1=7 one less than the number of terms.
A gentleman dying, left 9 sons, to whom and to his executors, he bequeathed his estate in manner following : To his executor £:50; his youngest son was to have as much more as the executor, and each son to exceed the next younger by as much more ; what was the eldest son's portion ?