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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Analysis and Approaches |
The sum of the first n terms of a geometric sequence is given by:
$$S_n = \sum_{r=1}^{n} \dfrac{3}{4}\left( \dfrac{5}{7} \right) ^r $$(a) Find the first term of the sequence, \(u_1\).
(b) Find \(S_\infty\).
(c) Find the least value of \(n\) such that \(S_\infty - S_n < 0.001\).
2. | IB Standard |
(a) Expand the following as the sum of six terms:
$$ \sum_{r=3}^{8} 2^r$$(b) Find the value of:
$$ \sum_{r=3}^{25} 2^r$$(c) Explain why the following cannot be evaluated:
$$ \sum_{r=3}^{\infty} 2^r$$3. | IB Studies |
Consider the number sequence where \(u_1=500, u_2=519, u_3=538\) and \(u_4=557\) etc.
(a) Find the value of \(u_{30}\)
(b) Find the sum of the first 12 terms of the sequence:
$$\sum_{n=1}^{12} u_n $$Another number sequence is defined where \(w_1=4, w_2=8, w_3=16\) and \(w_4=32\) etc.
(c) Find the exact value of \(w_{10}\).
(d) Find the sum of the first 9 terms of this sequence.
\(k\) is the smallest value of \(n\) for which \(w_n\) is greater than \(u_n\).
(e) Calculate the value of \(k\).
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