made, if upon 
mo- in the same 
nts of £l per 
annuity by the 
e nearest quan- 
; (by observing 
ximation to the 
in 10 years to 
aum in 10 years : 
e amounts of £l 
035 
.03 
. 005< 
difference 
of interest. 
1.375425 
276 — 11.6035 
suit is very near 
cent: if .03263 
; then have ob- 
= .03263, and 
r cent the propor- 
ntly near for most 
rte. 
nd interest is con- 
mmber of periods 
two payments of 
a year is therefore 
COMPOUND INTEREST. 37 
and the following series is therefore the amount of an annuity of ¿£'1 
at the expiration of n years, since each payment is 
H I+ ( l+ iy + 0 + i)-'+0 + ij z + •••• 
+ 1 + - 
7 \ m (. mn — 2 ) 
r 
+ ( 1 + 
Substituting, as in Art. 44, we have here 1 = the first term, min — 
the number of terms, and ( 1 + — ) m ' = the common ratio, and the 
\ m ) 
sum of the series will be 
/, i\ mn 
i 0 + -) - 
\ mj 
(l + - 1 
\ m/ 
1 + 
m 
- 1 
when m — in' then — — 1, and the formula becomes 
a ( l + =) - 1 
a l 1 + 
<6 \ on 
( 1 + ^) - 1 
What will an annuity of £20 amount to in 12 years at 6 per cent 
compound interest, when annuity and interest are payable half-yearly?