for the Valuation of Life-annuities. 
1 27 
i+r I + ,y l+r y I + ,p — ^ 
fent value of an annuity certain beginning with f. 1, 
and increafing to f. 2 the feconcl year, to f. 3 the third 
year, See. 
If this laft annuity is not an annuity certain for a 
given term, but a life- annuity, the value of it (fuppoling 
n the complement of life, a the value of an annuity 
certain for n years, g the value of two equal joint lives 
whofe common complement is n, p the perpetuity, and 
p the value of f. 1 to be received at the end of n years) 
will be A-Gx/z + / 2 x/)xP-AxPxi+r. 
Let the term be forty-one years, and the rate of inter eft 
4 per cent. 
The value of an annuity of f. 1 certain for this term 
is^f. 20. 
The value of an annuity certain for the fame term, 
and beginning with f. 1 at the end of the firft year, 
but increafing to f. 2 at the end of the fecond year, to 
f. 3 at the end of the third year, and fo on till it be- 
comes f. 4 1 at the end of the forty-firft year, is (by 
the fecond theorem, putting i+r, or 1,04 for a) 
f.2\\ 1 or. 
The value of an annuity increafing at this rate with- 
out end \i f. 650. 
EXAMPLES. 
If 
