The number of payments is 400. If M be the amount, we have A = 10044)™ ; -. log Mf = log 100 + 400 (log 81 — log 80) = 2 + 400 (4 log3 — 1 — 3 log2) = 2 + 400 (.0053952) = 4.15808 ;
whence M = 14390.6.
Thus the amount is $14390.60.
Note. At simple interest the amount is $600.
452. To find the present value and discount of a given sum due in a given time, allowing compound interest.
Let P be the given sum, V the present value, D the discount, R the amount of $1 for one year, n the number of years.
Since V is the sum which, put out to interest at the present time, will in 2 years amount to P, we have
P= VR^n V= PR^{-n} and D=P-V=PA—&*).
ANNUITIES.
453. An annuity is a fixed sum paid periodically under certain stated conditions; the payment may be made either once a year or at more frequent intervals. Unless it is otherwise stated, we shall suppose the payments annual.
454. To find the amount of an annuity left unpaid for a given number of years, allowing compound interest.
Let A be the annuity, R the amount of $1 for one year, n the number of years, M the amount.
At the end of the first year A is due, and the amount of this sum in the remaining n-1 years is AR^{n-1} at the end of the second year another A is due, and the amount of this sum in the remaining n-2 years is AR^{n-2}; and so on.
o M= AR^{n-1} + AR^{n-2} +... + AR + AR + A
=A(1+R+ R^2+..to n terms) =A (R^n-1) (R^n-1)
