38 ON THE VALUE OF ANNUITIES. 
We may also find the amount by multiplying by 10 the amount of ¿£1 
per annum in 24 years at 3 per cent. 
34.4264*7 
10 • 
344.2647 = <£344 5 3. 
PRESENT VALUES OF ANNUITIES AT COMPOUND INTEREST. 
49. The present value of £l payable at the end of one year, (Art. 35.) 
is (1 + 0“ 1 , at the end of two years, (1 + 0 \ and generally at the 
end of n years (1 + i)~ n ; and the present value of an annuity being 
equal to the aggregate of the present values of the several payments, 
the following series will be the present value of <£l per annum for 
71 years: 
(i + O’ 1 + (1 + 0~ 2 + (1 + 0~ 3 + (1 + o~ 4 + — 
.... + (1 + 0~ (n ~ 2) + (1 + 0" (n_1) + (1 +i)- n ; 
the first term of which is (1 —j- i)~ l , the common ratio (1 + i)~ and 
the number of terms n; the sum of the series by the formula ■ ° 
1 — r 
(Arith. and Alg. 115.), where a denotes the first term, n the number 
of terras, and r the common ratio will be found equal to 
. 1 _L j\- i _ (i + o» + o 
v : , which becomes, by multiplying nume- 
1 _ (i + 0" 1 J 1 J G 
rator and denominator by (1 + 0, 
1 - (1 + i)- n ] 
i - (i + o- 
(1 + 0 
Let us now make 
1 
p == present value, 
a — annuity, 
7i — number of years, 
i = interest of ¿£l for one year. 
50. To find (p) the present value— 
Multiplying the present value of <£l per annum just found by a, we 
have the present value of £ a per annum 
1 - (1 + i)~ n ‘ 
p ~ a. ; . 
i 
Rule. Subtract from unity the present value of £l due at the expi 
ration of the number of years the annuity has to continue, and divide 
the difference by the interest of £l for one year; the quotient multi 
plied by the annuity gives its present value. 
Example. A holds for the term of 20 years an estate by lease, of the 
value of £250 per annum, for which he pays an annual rent of £80.