Let p — the premium for ¿£l for the first t years, p t for the next t, 
years, for the succeeding t t , years, &c., and call the last premium 
commencing after the payment of v premiums P ; then, supposing the 
last premium to have been just paid, and t more premiums of j> each to 
be payable before any variation takes place, and the age at the time of 
valuation to be m, the present value of all the future premiums will be 
p.«0) +P/*«(m) + 2V«(m) + +P.a ( m) , 
n Qt Ut+t, 
which subtracted from the single premium for the assurance at the 
present age (m), Avill give the value of the policy, viz., 
A m — +P ; Ct( m ) +P//.« (m) _ + + P«(m) ) > 
1« 
which may be thus written ; 
M m -{p(N m -N,„ +f )+7J / (N m+i -N m+t+f/ )4-^ // (N w , +t+(| -N m+t+f|+ti/ )+..+P.N, n+t } 
If the last premium be just due and not paid, i.e., if there be (¿+1) 
premiums of p each to be paid (one of them immediately), the value 
will be 
Am — { V (1 + «(,n) ) + P/«(m) "f P« • «(«) __ + + P • «(m) }, 
(1 <7i i 1! ' 
which may be thus written : 
M m -p(N m . 1 -N w+t )+p / (N m+f -N w+t+t< )+p // (N m+<+ i < -N w+ , +tj+ ^+...+P.N wl+ , 
D„ 
INCREASING AND DECREASING ANNUITIES. 
262. To find the value of an increasing annuity, 
Let there be n perpetuities of £l per annum, the first to be entered 
on immediately, the second at the end of one year, the third at the end 
of two years, and so on to the wth, which is to be entered upon at the 
end of n— 1 years. By Art. 56, the present value of the first per 
il+0' 
of the third, 
&c., and 
; the present value of the n perpetuities will 
therefore be 
l + (l + 0~ l +(l+i)- 2 +(l-W)" 8 + (l+0 -( ” _1) 
i ’ 
the numerator of this expression is unity, added to the present value of 
£\ per annum for n — 1 years, (Art. 49) ; the value of the series is 
therefore 
(i+o-a+o-^i 
if from this we subtract the present value of n perpetuities, each of £l 
per annum deferred n years, we shall have remaining the value of an