198 
LIFE ANNUITIES. 
264. Let a person aged m be entitled to n annuities of £l each, 
payable until his decease, the first to be entered upon immediately, the 
second at the end of one year, the third at the end of two years, and so 
on to the wth, which will be entered upon at the expiration of tz— 1 
N 
years; the present value of the first will be ~, the present value of 
J-'777, 
the second 
N„ 
, the present value of the ?ith, 
N, 
m-f-jz-1 
the value of 
the n annuities will be 
N m+1 + N m+2 +N m + 3 4" • • • • • a — j 
D m 
Since column S, opposite each year of age gives the sum of the 
numbers in column N at each age and at all ages above—if, therefore, 
from the number in column S opposite to the age m we subtract the 
number in the same column opposite the age m + n, we have the sum 
of the first n terms in column N ; the expression just obtained is 
therefore 
^m-fn 
‘ 
If all payments cease at the end of n years from this time, the 
present value of each annuity will be diminished by the present value 
of a life annuity to be entered upon at the expiration of n years, viz., 
IX 
subtracting 
n. N, 
77l-f-7l 
D„ 
s„ 
from 
S m -S„ 
D„ 
-, we have 
■S m4 .„—n. N, 
IX 
the present value of an 
annuity for n years, the first payment being £l, and increasing by ¿£'1 
annually until the end of the term. 
If we multiply by p, we have 
P (S,» S m .j. n m ) 1 j, 
x— s the present value of an 
Ufa 
annuity for n years, commencing at £p, and increasing £p annually. 
If the first payment be £a, and the future payments be increased 
annually by £p, we must add the present value of an annuity of a — p 
pounds for n years, viz., 
(a~p)(N m -N ra+n ) , . , . 
D » which gives 
(a - p) (N m - N m+ „) + p(S m - S m+n - ti ■ N m+ „) 
265. If instead of p we take — p, the expression becomes 
(a+X)( N m-N m +„)—p(S m — S m+n —7i.N, n+n ) 
Dm 
the present value of an annuity for n years, commencing at £a, and