# Exam-Style Questions.

## Problems adapted from questions set for previous Mathematics exams.

### 1.

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:

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

### 2.

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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