The volume of water VnV_nVn (in megalitres) in a reservoir at the end of year n n\,n is modeled by the recurrence relation Vn+1=kVn+50V_{n+1} = kV_n + 50Vn+1=kVn+50, where k k\,k is a constant and V1=200V_1 = 200V1=200.
Find an expression, in terms of kkk, for the volume V2V_2V2.
It is given that the volume at the end of the third year is V3=152V_3 = 152V3=152 megalitres.
Show that k k\,k satisfies the equation 100k2+25k−51=0100k^2 + 25k - 51 = 0100k2+25k−51=0.
Given that the volume of the reservoir is strictly decreasing year-on-year, find the value of V4 V_4\,V4 and the value of V5V_5V5.
The volume of water in the reservoir approaches a limit L L\,L as n→∞n \rightarrow \inftyn→∞.
Write down an equation for LLL.
Find the value of LLL.
