7

Ratios, AND PROPORTION. (16.) Ratios. Principles.

If a >b, then

(E. 440-50.)

a + x : 6+ x > or <a : b, according as a < or > b.

(W. 157—70.)

and generally, ma + nb : pa + qb :: mc+nd : pc+qd.

(W. 171-182; E. 461-5.) If a : 6 :: C:d :: e: f

:: &c. then
a : 6 :: a+c+e+ &c : 6+ d +f+ &c.

d
ma : mb ::

;

:

n

n

If a : 6 :: C : d,

and e:f:: g: h; then
ae : bf :: cg : dh.

(W. 184–8.) If two numbers be prime to each other they are the least in that proportion.

(W. 189.) Compound proportion.

(E. 488-504.)

ARITHMETICAL PROGRESSION.

(18.) Let am represent the moth term of an arithmetical series,

b, the common difference,
8, the sum of n terms;

To insert n arithmetical means between c, and e.

Given the pth and oth terms, to find the oth term.

-р
0,
= a + (ag-a)

9-P
(W. 212; E. 402-24; Bour. 1938.)

(19.) Any three of the quantities az, an, b, n, s; being given, the others may be found from the following formulæ :

HARMONICAL PROGRESSION. (20.) The reciprocals of quantities in arithmetical progression are in harmonical progression, and conversely. To insert n harmonical means between c, and e. The mth mean =

(n+1) ce

(n-m + 1)e+mc Given the pth and qth terms, to find the poth term. (q-pap.aq

(W. 381.) (r-pa,-(-9)

FIGURATE, AND POLYGONAL NUMBERS.

(21.) The nth term of the mth order of figurate numbers = the sum of n terms of the (m- 1)th order

n 7-min + m 1......N+1.
1
2

m .m +1

n

a

« The above proposition may be thus compendiously stated:

For an explanation of this notation see the Appendix.

The nth term of the series of m-gonal numbers is the sum of n terms of the arithmetical progression

1, 1+ m - 2, 1 + 2.m-2, &c.
1
{m–2.no— m

— m – 4. n}. The sum of the reciprocals of n figurate numbers of the mth order

1.2.3...m 1

1 m-1 1.2.3...(m-1) (n+1) (n + 2)...(n + m - 1)

(G. A. 132, 3; L. C. 91, 2; E. 425–39; B. 27, 8.)

)}

On-m: @ne + m=(an) .
To insert n geometrical means between c and e.

m

n+1

The mth mean=c

°C)
Given the pth and oth terms, to find the path term.

(W. 214–22; E. 505-22; Bour. 199 208.)