Sequences

Algebra

Questions marked Higher are aimed at the Higher tier.

These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.

Find the next two terms of $3, 7, 11, 15, \ldots$

2 marks
Find the $n$th term of $5, 8, 11, 14, \ldots$

2 marks
Find the $n$th term of $2, 6, 10, 14, \ldots$

3 marks
The $n$th term of a sequence is $3n + 2$. Is $100$ a term in the sequence?

3 marks
The $n$th term of a sequence is $n^{2} + 1$. Write the first three terms and the $10$th term.

4 marks
A sequence has $n$th term $5n - 3$. Find the first term, and find which term equals $97$.

4 marks
Find the $n$th term of $2, 6, 12, 20, 30, \ldots$ Higher

5 marks
A geometric sequence starts $3, 6, 12, 24, \ldots$ Find the next term, the $n$th term, the term that equals $768$, and the sum of the first five terms. Higher

6 marks

Mark scheme

Answer. $19$ and $23$
- The common difference is $4$.
- $19$ and $23$
Answer. $3n + 2$
- The difference is $3$, and the term before the first would be $2$.
- $3n + 2$
Answer. $4n - 2$
- The difference is $4$.
- $4n - 2$
Answer. No
- $3n + 2 = 100$ gives $3n = 98$.
- $98$ is not divisible by $3$, so $100$ is not a term.
Answer. $2, 5, 10$ and $101$
- $1 + 1 = 2$, $4 + 1 = 5$, $9 + 1 = 10$.
- $100 + 1 = 101$
Answer. First term $2$, and it is the $20$th term.
- When $n = 1$, the term is $2$.
- $5n - 3 = 97$ gives $n = 20$.
Answer. $n^{2} + n$
- The second differences are constant and equal to $2$, so the sequence is quadratic.
- Compare with $n^{2}$: the sequence is $n^{2} + n$.
- Check: the third term is $9 + 3 = 12$.
Answer. Next term $48$, $n$th term $3 \times 2^{n-1}$, the $9$th term, sum $93$
- Each term doubles, so the next term is $48$.
- The $n$th term is $3 \times 2^{n-1}$.
- $3 \times 2^{n-1} = 768$ gives $2^{n-1} = 256 = 2^{8}$, so $n = 9$.
- The first five terms sum to $3 + 6 + 12 + 24 + 48 = 93$.