20 
ON THE VALUE OF ANNUITIES. 
By transposition, 
(i + o 
As this series converges very fast we shall he sufficiently accurate if 
we omit all the terms hut the first two ; we then have the equation 
Multiply hy 6 m", and dividing by b 
3 iri 1 — 3 m -f imP — 3 bn -f 2 i = 6 nv. 
) 
s 
- - (i + 0 
Hi 2 {3 -f i — G — 
— (3 + 31) in — — 2 i. 
This equation is a quadratic, and if we substitute for i, s, and p their 
values in figures, and solve the equation (Arith. & Alg. 206) we shall 
have the value of m very nearly. 
32. To find (i) the rate of interest 
(Art. 24.) 
dividing each side hy p, 
extracting the 7n?ith root of each side, 
hy transposition, 
Multiply each side hy m: 
— j mn is found hy logarithms : log 
Rule. Divide the difference between the logarithms of the amount 
and principal, by the product of the number of years and of periods of 
conversion of interest in a year; the quotient is a logarithm; find the 
corresponding number, and from it subtract one, and multiply the differ 
ence by the number of periods interest is convertible in a year. 
Example, At what rate per cent will £210 6 0 at compound inte 
rest payable quarterly, amount to £690 in 15 years? ,