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PILING OF BALLS AND SHELLS.

(192.) Balls and shells are usually piled in three different forms, called triangular, square, or rectangular, according as the figure on which the pile rests is triangular, square, or rectangular.

(1.) A triangular pile is formed by continued horizontal courses of balls or shells laid one above another, and these courses or rows are usually equilateral triangles whose sides decrease by unity from the bottom to the top row, which is composed simply of one shot; and hence the series of balls composing a triangular pile is

1+3+6+10+15+.

n(n+1),

......

where n denotes the number of courses in the pile.

2

(2.) A square pile is formed by continued horizontal courses of shot laid one above another, and these courses are squares whose sides decrease by unity from the bottom to the top row, which is also composed simply of one shot; and hence the series of balls composing a square pile is

1+4+9+16+25+ . . . . n2,

where n denotes the number of courses in the pile.

(3.) The rectangular pile may be conceived to be formed from a square pile, by laying successively on one face of the pyramid a series of triangular strata, each consisting of as many balls as the face itself contains, and the number of these added triangular strata is always one less than the number of shot in the top row; therefore, if ʼn denote the number of courses, and m+1 the number of shot in the top row, the series composing a rectangular pile is (m+1)+2(m+2)+3(m+3)+4(m+4)+ ....n(m+n) =m+2m+3m+4m+ nm+12+22+32+42+ . =m(1+2+3+4+....n)+ square pile

n

....

_n(n+1),m+ square pile.

2

(4.) The number of balls in a complete triangular or square pile must evidently depend on the number of courses or rows; and the number of balls in a complete rectangular pile depends on the number of courses, and also on the number of shot in the top row, or the amount of shot in the latter pile depends on the length and breadth of the bottom row; for the number of courses is equal to the number of shot in the breadth of the bottom row of the pile. Therefore, the number of shot in a triangular or square pile is a function of n, and the number of shot in a rectangular pile is a function of n and m.

(5.) If the general term of any series of numbers be of the mth degree, the sum of all the terms of such series will be of the (m+1)th degree; because

the general term of any progressively increasing series being a function of n of the mth degree, the sum of such series evidently cannot exceed n times the general term, that is, it cannot exceed n times a function of n of the mth degree, and therefore the function itself must of the (m+1)th degree.

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Here n, the general term of the series, is of the first degree, and therefore the function expressing the sum of the series is of the second degree; and hence we assume

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Now this equation must be true for every value of n; hence, wher.

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(2.) Sum n terms of the series 1+3+6+10+15+.

2

n(n+1).

....

2

Assume 1+3+6+10+15+

...

n(n+1)=Pn3+Qn2+Rn+S..........

2

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and since there are four coefficients to be determined, we must have a corresponding number of independent equations; hence

when n=1 we have

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PILING OF BALLS AND SHELLS.

(192.) Balls and shells are usually piled in three different forms, called triangular, square, or rectangular, according as the figure on which the pile rests is triangular, square, or rectangular.

(1.) A triangular pile is formed by continued horizontal courses of balls or shells laid one above another, and these courses or rows are usually equilateral triangles whose sides decrease by unity from the bottom to the top row, which is composed simply of one shot; and hence the series of balls composing a triangular pile is

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where n denotes the number of courses in the pile.

2

(2.) A square pile is formed by continued horizontal courses of shot laid one above another, and these courses are squares whose sides decrease by unity from the bottom to the top row, which is also composed simply of one shot; and hence the series of balls composing a square pile is

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where n denotes the number of courses in the pile.

(3.) The rectangular pile may be conceived to be formed from a square pile, by laying successively on one face of the pyramid a series of triangular strata, each consisting of as many balls as the face itself contains, and the number of these added triangular strata is always one less than the number of shot in the top row; therefore, if n denote the number of courses, and m+1 the number of shot in the top row, the series composing a rectangular pile is (m+1)+2(m+2)+3(m+3)+4(m+4)+ .... n(m+n)

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(4.) The number of balls in a complete triangular or square pile must evidently depend on the number of courses or rows; and the number of balls in á complete rectangular pile depends on the number of courses, and also on the number of shot in the top row, or the amount of shot in the latter pile depends on the length and breadth of the bottom row; for the number of courses is equal to the number of shot in the breadth of the bottom row of the pile. Therefore, the number of shot in a triangular or square pile is a function of n, and the number of shot in a rectangular pile is a function of n and m.

(5.) If the general term of any series of numbers be of the mth degree, the sum of all the terms of such series will be of the (m+1)th degree; because

the general term of any progressively increasing series being a function of n of the mth degree, the sum of such series evidently cannot exceed n times the general term, that is, it cannot exceed n times a function of n of the mth degree, and therefore the function itself must of the (m+1)th degree.

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Here n, the general term of the series, is of the first degree, and therefore the function expressing the sum of the series is of the second degree; and hence we assume

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Now this equation must be true for every value of n; hence, wher.

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and since there are four coefficients to be determined, we must have a corresponding number of independent equations; hence

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Formula for a square pile.

(3.) Sum n terms of the series 1+4+9+16+25+ . . . . n2. Assume 1+4+9+16+25+ ... n2=Pn3+Qn2+Rn+S; then, as before, P+ Q+ R+S=1

8P+ 4Q+2R+S=1+4 27P+9Q+3R+S=1+4+9

= 1

= 5

=14

64P+16Q+4R+S=1+4+9+16=30

and from these four equations we find, by continued subtraction,

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(4.) By Art. 192, we have the number of shot in a rectangular pile

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64P+16Q+4R+S=(m+1)+2(m+2)+3(m+3)+4(m+4)=10m+30

and from these four equations we find, as before,

P=', Q='(m+1), R='(3m+1), and S=0; hence

(m+1)+2(m+2)+3(m+3)+.....n(m+n)='n3+m+1n2 + 3m +1n.

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6

= {2n2+3(m+1)n+3m+ }}

= {2n(n+1)+(n+1)+3m(n+1)}

_n(n+1) 2n+1+3m

2

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3

expressions for (S) the number of balls

_1 n(n+1). (n+1+1)

=

3 2

s="(n+1)(2n+1)= n(n+1),(n+1+n)

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3

rectangular, S="(n+1)(2n+1+3m)=1,n(n+1){(n+m)+(m+1)+(n+m)}

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