Exam-Style Questions.

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 \(S_\infty - S_n < 0.001\).

Worked Solution

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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:

$$ \sum_{r=3}^{\infty} 2^r$$

Worked Solution

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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 \(k\).

Worked Solution

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