Exam-Style Questions.

1.	GCSE Higher

(a) Find the \(n\)th term of the sequence 7, 13, 19, 25,...

(b) In a sequence of four numbers, the difference between each number is 9.
The sum of the four numbers is 2.
What are the numbers in the sequence?
You must show all your working.

Worked Solution

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2.	IB Standard

The first three and last terms of an arithmetic sequence are \(7,13,19,...,1357\)

(a) Find the common difference.

(b) Find the number of terms in the sequence.

(c) What is the sum of the sequence.

Worked Solution

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3.	IB Standard

An arithmetic sequence is given by 6, 13, 20, …

(a) Write down the value of the common difference, d.

(b) Find \(u_{100}\);

(c) Find \(S_{100}\);

(d) Given that \(u_n=1434\) , find the value of n.

Worked Solution

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4.	IB Standard

In an arithmetic sequence, the fifth term is 44 and the ninth term is 80.

(a) Find the common difference.

(b) Find the first term.

(c) Find the sum of the first 50 terms of the sequence.

Worked Solution

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5.	IB Applications and Interpretation

A celebrity football match is planned to take place in a large stadium.

The most expensive tickets are in the first row. The ticket price for each row forms an arithmetic sequence. Prices for the first four rows are shown in the following table.

Row number:	1	2	3	4
Price per seat	£50	£48.50	£47	£45.50

(a) Write down the value of the common difference, \(d\).

(b) Calculate the price of a ticket in the 19th row.

(c) Find the total cost of buying 5 tickets in each of the first 10 rows.

Worked Solution

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6.	IB Analysis and Approaches

An arithmetic sequence has first term 99 and common difference \(-5.5\).

(a) Given that the \(k\)th term of the sequence is zero, find the value of \(k\).

Let \(S_n\) denote the sum of the first \(n\) terms of the sequence.

(b) Find the maximum value of \(S_n\).

Worked Solution

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7.	IB Studies

A Grecian amphitheatre was built in the form of a horseshoe and has 22 rows.

The number of seats in each row increase by a fixed amount, \(d\), compared to the number of seats in the previous row. The number of seats in the fifth row, \(u_5\), is 58, and the number of seats in the ninth row, \(u_{9}\), is 86. \(u_1\) represents the number of seats in the first row.

(a) Write an equation for \(u_5\) in terms of \(d\) and \(u_1\).

(b) Write an equation for \(u_{9}\) in terms of \(d\) and \(u_1\).

(c) Write down the value of \(d\);

(d) Write down the value of \(u_1\).

(e) Find the total number of seats in the amphitheatre.

Some time later, a second level was added to increase the amphitheatre’s capacity by another 2590 seats. Each row has five more seats than the previous row. The first row on this level has 82 seats.

(f) Find the number of rows on the second level of the amphitheatre.

Worked Solution

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8.	IB Analysis and Approaches

Consider an arithmetic sequence where \(u_{10}=S_{10}=15\). Find the value of the first term, \(u_1\) and the value of the common difference, \(d\).

Worked Solution

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9.	IB Analysis and Approaches

Consider the arithmetic sequence \( u_1, u_2, u_3, \ldots \).

The sum of the first \( n \) terms of this sequence is given by \( S_n = 2n^2 + n \).

(a) Find the sum of the first six terms.

(b) Given that \( S_7 = 105 \), find \( u_7 \).

(c) Find \( u_1 \).

(d) Hence or otherwise, write an expression for \( u_n \) in terms of \( n \).

Consider a geometric sequence, \( v_n \), where \( v_2 = u_1 \) and \( v_4 = u_7 \).

(e) Find the possible values of the common ratio, \( r \).

(f) Given that \( v_{30} < 0 \), find the value of \( v_5 \).

Worked Solution

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10.	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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11.	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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