3.4.2 Sequences and Series (A-level only)

The depth of water in a tidal sensor tank, $d_n$ mm, measured at the start of each hour $n$ , is modeled by the recurrence relation

d_{n+1} = P - Q d_n

where $P$ and $Q$ are constants and $d_1 = 6$ .

Find expressions in terms of $P$ and $Q$ for (i) $d_2$ (ii) $d_3$

[2]

Given that $∑n=13dn=60\sum_{n=1}^{3} d_n = 60$ and $P = 5 Q + 8$

Show that $Q^2 - 4Q - 38 = 0$ .

[4]

Hence find the smaller possible value of $d_2$ , giving your answer in the form $\sqrt{b}$ where $a$ and $b$ are integers.

[4]

3.4.2 Sequences and Series (A-level only) Questions