Edexcel C2 2006 June — Question 9

Exam BoardEdexcel
ModuleC2 (Core Mathematics 2)
Year2006
SessionJune
TopicGeometric Sequences and Series
  1. A geometric series has first term \(a\) and common ratio \(r\). The second term of the series is 4 and the sum to infinity of the series is 25.
    1. Show that \(25 r ^ { 2 } - 25 r + 4 = 0\).
    2. Find the two possible values of \(r\).
    3. Find the corresponding two possible values of \(a\).
    4. Show that the sum, \(S _ { n }\), of the first \(n\) terms of the series is given by
    $$S _ { n } = 25 \left( 1 - r ^ { n } \right) .$$ Given that \(r\) takes the larger of its two possible values,
  2. find the smallest value of \(n\) for which \(S _ { n }\) exceeds 24 .
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