INCREASING AND DECREASING SCALE OF PREMIUMS. 197 
annuity for n years, commencing with £l, and increasing each 
year, viz.: 
1 f + l w(l + t)~" _ 1 f 1 + i— (1 + 
»1 » ' i ~i l 
. P(l + i-(l + *)“ (n " 1) 
5 ) 
t{ 
■n(l+i) — the value of an annuity for 
i t i 
years, whereof the first payment is £p, the second £2p, the third £3p, 
increasing £p each payment until the expiration of the annuity. 
263. If the first payment be £a, and the future payments be in 
creased by £p each year, we must add to the value just found the 
present value of an annuity of a—p pounds for n years : 
N i-(i+o _n , v ii+*— (i+0" (n " i) , ] 
(a-p) 7 + T n (l+z)-"j = 
a. 
i i ( % 
(i+0" n . pfi—(i+0“ < "" ,> 
+p. 
+ 
ft 
■w(l +0 -B | = 
l-(l + *) _ " , ip(l+0- B +p{l-(l-fi)- Cn - 1 ^iVi(l+i)-n 
a. : i r ■- 
l l 2 
1 —(1 + 0"" . { (i—+1 —(l + i) (l+0 -n l 
a. +p * ~ 
1— (l + i) — 
+ p{“ 
(1 + m) (l + z)~ n l 
If p be changed in sign, the decreasing annuity is 
1—(1 +i)-” _ 1—(l+*n)(l+*)- 
a. 
-v 
i v- 
Example. What is the present value of an annuity for the next 
10 years commencing at £20, and increasing ¿£10 each year, at 4 per 
cent compound interest? 
1. 
1.04~ 10 = .67556417 
.04).32443583 
8.110896 
02 
.04 
.10 
in~ .40 
1 
1.4= I + in 
162.218 
.67556417 
4.1 
67556417 
27022567 
(1+*»)(!+i)" B = -94578984 
1. 
1 — (1 + in) (1 + i)~ v — . 05421016' 
10 
. 04).5421016, 
.04)13.55254 
338.813 
162.218+338.813=501.031 = ^501 0 8.