Compounded semiannually. P dollars is invested at annual
interest rate r for 1 year. If the interest is compounded
semiannually, then the polynomial P(1+r/2)^2represents the
value of the investment after 1 year. Rewrite this expression
without parentheses. Evaluate the polynomial if
P =$200 and r =10%.
it is totally impractival to write P(1+r/n)^n without brackets, unless n=2
that would be P(1 + r + r^2/4)
= P + Pr + Pr^2/4
amount = 200(1.05)^2
= 220.50
notice in the expanded version
amount = 200 + 200(.10) + 200(.10)^2/4
= 200 + 20 + 0.5
= 220.50
mmmh, compare the two calculations.
7300@7% COMPOUNDED SEMIANNUALLY FOR 3 YRS
To rewrite the expression without parentheses, we can use the exponent property that states (a^m)^n = a^(m*n). In this case, we have (1+r/2)^2. Distributing the exponent 2 to both terms inside the parentheses gives us (1+r/2)(1+r/2).
To multiply these two binomials together, we can use the FOIL method (First, Outer, Inner, Last).
First: Multiply the first terms of each binomial. (1*r/2 = r/2)
Outer: Multiply the outer terms of each binomial. (1 * 1 = 1)
Inner: Multiply the inner terms of each binomial. (r/2 * 1 = r/2)
Last: Multiply the last terms of each binomial. (r/2 * r/2 = r^2/4)
Putting it all together, we have:
(1+r/2)(1+r/2) = 1 + r/2 + r/2 + r^2/4 = 1 + r + r^2/4
Now, we can evaluate the polynomial if P = $200 and r = 10%.
Substituting the values into the polynomial:
P(1+r/2)^2 = 200(1+0.10/2)^2 = 200(1+0.05)^2 = 200(1.05)^2
To compute this expression, we can square 1.05 and then multiply it by 200.
1.05^2 = 1.1025
200 * 1.1025 = 220.50
Therefore, the value of the investment after 1 year would be $220.50.