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.
Practise AQA A Level Maths 1.7 D: Sequences and series with exam-style questions for A Level Maths. 267 questions covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series in modelling (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
