208 
LIFE ASSURANCES. 
diminishing annually by £ -; by the formula of Art. 265, this 
becomes 
1+ 
•+■ ~(N, n —(S t (n l) N„ 1+n _ 1 ) 
lit IX J Tt 
11 — 1 1 
but 1— — 1, the expression will therefore become 
n n 1 
Dm+N m —N m+n _! — — (S m — N m+il _i) + N m+ „_J 
D„ 
Dm + N m — - { S m — (S m+n _i- 
N„ 1+n _j) } 
D„ 
n 
-S m+n ) 
Dm 
since, by the construction of the tables, + N m — N„,_i, and 
M„ t 
S m+n _i—N m+B _i=S^; and since ^r— is the single premium for the 
D'm 
assurance of £l, by dividing by the expression just found, we have 
—— , the first premium to be required for the assur- 
N m _i — (S„ t S,„ + „) 
ance of £l on the life, supposing the subsequent payments to be suc 
cessively reduced by the nth part of the first premium, until they alto 
gether cease after n payments. 
What annual premium should be charged for the assurance of £100 
on a life aged 40, the premiums being successively reduced by the tenth 
part of the first premium, and ceasing altogether after the tenth pay 
ment ? (Northampton 3 per cent.) 
S, =S«=209130.1 N„ 
S m+n = S i0 = 85391.6 
10)123738.5 
12373.85 
:N 39 = 17659.528 
12373.85 
5285.68 )599.9792(.11352 = 
528568 £11 7 0 
i 714112 per cen ^. 
528568 
185544 
158570 
26974 
26428 
546