TO THE REV. DR. BARNES.

SIR,

I BEG your acceptance of the following paper;

and if

you

think that it deferves the attention of your Philofophical Society at Manchester, I take the liberty of requesting you to prefent it to that learned body. I have the honour to be,

SIR,

Your most obedient humble Servant,

TURNHAM GREEN,

JOHN ROTHERAM.

MIDDLESEX, Dec. ». 1787.

Some PROPERTIES of GEOMETRICAL SERIES explained in the SOLUTION of a PROBLEM, which By JOHN bath been thought indeterminate.

ROTHERAM, M. D.

PROBLEM.

Given, the fum (a), and the fum of the fquares (b), of any Geometrical Series: to determine the Series.

SINCE

INCE every Geometrical Series, as, x xr xr2 xr3 ..... to xr-1 is universally expresfed by three quantities, viz. x, the first term,

r, the

r, the common ratio, and n, the number of terms; and fince there are only two conditions, given by the problem, whence these three quantities are to be found, it hath hitherto been thought indeterminate, or one that would admit infinitely many answers.

In an infinite geometrical feries, whose sum, x+xr+xr2+xr3, &c. ad infinitum = a, is a finite quantity, it is evident that r, the common ratio, must be less than unity: but it may be either a pofitive or a negative quantity. If it be a pofitive quantity, then it must be a proper fraction; but if negative, it may be either a proper fraction, or an improper fraction, or a whole number. If r be pofitive, the feries will be, x+xr+xr2+xr3, &c. ad infinitum; but if negative, it will be x-xr+xr-xr3, &c. ad infinitum. I fhall firft folve the general cafe where x+xr+xr2+xr3,,&c. ad infinit, a, and x2+x2r2+x2r°+x2p3, &c. ad infinit. = b.

2

2

2

=

If the first of these equations be multiplied by r, it will be xr+xr+xr3 +xr+, &c. ad inf.ar; and this, taken from the firft, leaves

x=a-ar.

If the fecond equation be multiplied by r2, it will be x2r2+x2ra + x2 μ ®, &c. ad inf.=br, and this, taken from x2 +x2j2 +x2 y4, &c. ad inf.=b, leaves x2-bby 2.

2

But

But x-a-ar, whence x2-a2. · 2 a2r+a2r2; and confequently, b-bra2 - 2 a2r+a2r2.

Whence, after proper reduction, r

a2-b

a2+b°

This value of r being fubftituted in the equa. tion x-a-ar, gives, after proper reduction,

x=

2 ab a2+b2

Whence, and being found, the

feries will be known.

EXAMPLE. Let a=24, and b-192; then

a2 —b_576—192_384

=

a2 + b 576+ 192 768

24×192×2

768

2ab

; and x=.

a2 + b

12; and the feries is 12, 6, 3, 14,

3 , &c. ad inf.

Again. Let a=243, and b=295244; then r=+, and x 162; and the feries will be 162,

=

2

2

54, 18, 6, 2, 3, 3, 27, &c. ad infinitum.

31

2

In the infinite feries x+xr+xr2+xr3, &c, ad inf.=a, a=, because, xa-ar, as above.

By a fimilar deduction, b=

reason, if xr" be the

x 2

I-r

For the fame

first term of an infinite feries (n being any pofitive whole number) then,

The finite feries x+xr+xr2+xr3 + to xr is the difference of two infinite feries, of which x is the first term of the greater, and xr2 of the lefs feries, and confequently the fum of

the

and if x be taken for the leaft term,

and xr for the greateft, in which cafer will

2

=b. From these two equations

only, it is impoffible to obtain the values of the three unknown quantities, x, r, and n; recourfe must therefore be had to another property of the feries.

4

b

.... to

Let there be given x+xr+xr2+xr3, &c. .... to xra; and x2+x2r2+x2r^, &c. xr2-2b, where x, r, and n are whole pofitive numbers. Then the latter of these equations divided by the former, gives x-xr+xr2 —, xr3 +xr*—xr3, &c. . If this be added to == the first equation, the fum will be, 2x+2xr2 +2xr, &c. =a+ ; and, if it be subtracted, the difference will be, 2xr+2xr3 +2xr3, &c. = 4. Again, let 2x+2xr2+2xr++2xr°, &c. a+be divided by 2xr+2xr3+2xr?, &c. a—, the quotiens will give r, and the re

b

a

mainder

mainder 2x; as will appear by the operation; 2xr+2xr3+2xr?, &c.) 2x+2xr2+2xr*+2xr°, &c. ( 2xr2+2x+2xr6, &c. *the re

r, in the quotient

2x

mainder.

Hence this general Rule.

Divide the fum of

the fquares by the fum of the feries.

quotient to, and fubtract it from,

Add the

the fum of the feries. Divide the greater of these two numbers by the lefs; the quotient of this fecond divifion fhall be the common ratio, and the remainder twice the first term.

EXAMPLE I. Let a 242, and b=29524. Then=122, and 242 +122=364, and 242– 122 120; and 364 divided by 120, gives 3 in the quotient, for the common ratio, and 4 in the remainder, the half of which, 2, for the first term of the series. And, these being known, the number of terms, will be found by the common rules. Whence the feries is 2, 6, 18, 54, 162.

EXAMPLE II. Let a=68887 and b=2372950489; then == 34447, and a += 103334, which di

a

b

a

vided by a- 34440, gives, in the quotient, 3 for the common ratio, and in the remainder 14, the half of which, 7, for the firft term; whence the feries is 7, 21, 63, 189, &c. to 9 places.

EXAMPLE