after the 
onary or 
ained by 
mediately 
btracting 
mly until 
s evident 
years, its 
;d during 
muity for 
tmues, 
years de 
ls the last 
e by the 
£.30 per 
the next 
19 0 84 
unity and 
sent value 
61. To find (n) the number of years. 
multiply by i and divide by a, 
(i + 0 “ d - (i + i) 
by transposition, (1 + i) ~ d 
multiply each side by (1 + i) d 
(1+0°- — (!+*)'= (1 + 0 
but (1 + ¿) 0 — 1j 
(i + *)"" = i — ~ (1 + 0 
by logarithms — n X log (I + i) ■= log {l — (1 + ?)'*} 
dividing each side by — log (1 + ¿), 
- log 11 — (i + O d | 
log (1 + 0 
Example. The sum of £119 0 8^ is given for the purchase of an 
annuity of £30 to be entered upon after the expiration of 10 years ; 
how long will the annuity continue, reckoning interest at 5 per cent ? 
p = 119.035 n — 10 i = .05 a = 30 
, w 119.035 X .05 X 1.05 10 
(I + i) d = 1 - 
Table 3, 1.05 10 — 1.628895.