a). Substitute 1 and 2 into the expression for R(n) to get two simultaneous equations for h and k:
n = 1: h1/2/k = 4.8 (working on millions of dollars)
n = 2: h1/k = 5.76 from this we have h = 5.76k. Put this into the first expression above to get:
(5.76k)1/2/k = 4.8 or 5.761/2/k1/2 = 4.8. Multiply both sides by k1/2 and divide both sides by 4.8 to get:
k1/2 = 5.761/2/4.8. Square both sides: k = 5.76/4.82
$${\mathtt{k}} = {\frac{{\mathtt{5.76}}}{{{\mathtt{4.8}}}^{{\mathtt{2}}}}} \Rightarrow {\mathtt{k}} = {\mathtt{0.25}}$$
Hence from the n = 2: equation above we have h = 5.76*0.25 or
$${\mathtt{h}} = {\mathtt{5.76}}{\mathtt{\,\times\,}}{\mathtt{0.25}} \Rightarrow {\mathtt{h}} = {\mathtt{1.44}}$$
Remember that the result of using these values in R(n) = hn/2/k will be in millions of dollars.
For the sum of R(n) to n, notice that it can be written as:
sum(n) = (a + a2 + a3 + ... + an)/k where a = h1/2
The numerator is a geometric progression giving the result of sum(n) = a*(an-1)/[(a-1)k]
or sum(n) = (h1/2)*(hn/2-1)/[(h1/2-1)k}
See if you can now have a go at part b).
.
a). Substitute 1 and 2 into the expression for R(n) to get two simultaneous equations for h and k:
n = 1: h1/2/k = 4.8 (working on millions of dollars)
n = 2: h1/k = 5.76 from this we have h = 5.76k. Put this into the first expression above to get:
(5.76k)1/2/k = 4.8 or 5.761/2/k1/2 = 4.8. Multiply both sides by k1/2 and divide both sides by 4.8 to get:
k1/2 = 5.761/2/4.8. Square both sides: k = 5.76/4.82
$${\mathtt{k}} = {\frac{{\mathtt{5.76}}}{{{\mathtt{4.8}}}^{{\mathtt{2}}}}} \Rightarrow {\mathtt{k}} = {\mathtt{0.25}}$$
Hence from the n = 2: equation above we have h = 5.76*0.25 or
$${\mathtt{h}} = {\mathtt{5.76}}{\mathtt{\,\times\,}}{\mathtt{0.25}} \Rightarrow {\mathtt{h}} = {\mathtt{1.44}}$$
Remember that the result of using these values in R(n) = hn/2/k will be in millions of dollars.
For the sum of R(n) to n, notice that it can be written as:
sum(n) = (a + a2 + a3 + ... + an)/k where a = h1/2
The numerator is a geometric progression giving the result of sum(n) = a*(an-1)/[(a-1)k]
or sum(n) = (h1/2)*(hn/2-1)/[(h1/2-1)k}
See if you can now have a go at part b).
.