C1 June 2007 Q8

EdexcelOld spec7 marksSequences & Series

8. A sequence \(a_1, a_2, a_3, \ldots\) is defined by\[\begin{aligned} a_1 &= k, \\ a_{n+1} &= 3a_n + 5, \qquad n \geqslant 1, \end{aligned}\]where \(k\) is a positive integer.

(a) Write down an expression for \(a_2\) in terms of \(k\). (1)
(b) Show that \(a_3 = 9k + 20\). (2)
(c)
(i) Find \(\displaystyle\sum_{r=1}^{4} a_r\) in terms of \(k\).
(ii) Show that \(\displaystyle\sum_{r=1}^{4} a_r\) is divisible by 10.
(4)
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