TMUA · Paper 1: Applications of Mathematical Knowledge

Arithmetic and geometric sequences

18 practice questions, each with the answer and an explanation.

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An arithmetic sequence has third term $11$ and seventh term $27$ . What is the $20$ th term?
What is the sum of the first $20$ terms of the arithmetic sequence $3, 7, 11, \dots$ ?
A geometric sequence has first term $5$ and common ratio $2$ . What is the sum of its first $8$ terms?
The geometric series $8 + 4 + 2 + 1 + \dots$ continues indefinitely. What is its sum to infinity?
A geometric sequence of positive terms has second term $6$ and fourth term $54$ . What is the first term?
What is the value of $3 + 5 + 7 + \dots + 21$ ?
How many terms are there in the sequence $7, 10, 13, \dots, 100$ ?
What is the coefficient of $x^{3}$ in the expansion of $(2 + x)^{5}$ ?
What is the coefficient of $x^{2}$ in the expansion of $(1 - 2 x)^{6}$ ?
What is the term independent of $x$ in the expansion of $(x + \frac{1}{x})^{6}$ ?
In the expansion of $(1 + x)^{n}$ , the coefficient of $x^{2}$ is $45$ . What is $n$ ?
What is the sum of all the integers from $1$ to $100$ inclusive that are not multiples of $3$ ?
A sequence is defined by $u_{1} = 3$ and $u_{n + 1} = 2 u_{n} - 1$ . What is $u_{6}$ ?
The sum of the first $n$ terms of a sequence is $S_{n} = 3 n^{2} + 2 n$ . What is the $10$ th term?
A geometric sequence has second term $12$ and fifth term $96$ . What is the sum of its first five terms?
The sum to infinity of a geometric series with first term $6$ is $9$ . What is the second term?
The first term of an arithmetic sequence is $5$ and the sum of its first $10$ terms is $185$ . What is the common difference?
What is the largest integer $n$ for which $1 + 2 + 3 + \dots + n < 500$ ?

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Arithmetic and geometric sequences

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