The volume of water, in litres, in a storage tank decreases by $10\%$ each day. However, $90$ litres

Question

The volume of water, in litres, in a storage tank decreases by $10\%$ each day. However, $90$ litres of water is also pumped into the tank at the end of each day. The volume of water in the tank at the end of $n$ days is denoted by ${{x}_{n}}$ and ${{x}_{0}}$ is the initial volume of water in the tank. Write down a recurrence relation to represent the above situation. [1] Given that $a{{r}^{n-1}}+...+a{{r}^{2}}+ar+a=a\left( \frac{1-r{{\,}^{n}}}{1-r} \right)$, Show that ${{x}_{n}}={{0.9}^{n}}\left( {{x}_{0}}-900 \right)+900$. [3] Deduce the limit of ${{x}_{n}}$ as $n$ becomes very large. [1]

Accepted Answer

$x_n = 0.9x_{n-1} + 90$; $x_n = 0.9^n(x_0 - 900) + 900$; $\lim_{n\to\infty} x_n = 900$

2008 MJC P1 Q8

2008 MJC P1 Q8

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