Worksheets · Foundation and Higher

Term rules, special and geometric sequences

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 4 marks
Here are the first four terms of a sequence: $5, 9, 13, 17$ .
(a) Write down the next term of the sequence. (1)
(b) Which is the term-to-term rule? (1)
1. Add 4
2. Multiply by 4
3. Add 5
(c) Is 50 a term in this sequence? You must show how you get your answer. (2)
1. Yes
2. No
Question 2Non-calculator · 5 marks
A sequence starts with 3. Each new term is found by multiplying the previous term by 2 and adding 3.
(a) Find the fourth term. (3)
(b) Find the term immediately before the first term if the same rule is used. (2)
Question 3Non-calculator · 5 marks
The $n$ th triangular number is $T_n = \dfrac{n(n + 1)}{2}$ .
(a) Show that the sum of two consecutive triangular numbers, $T_n + T_{n + 1}$ , is always a square number. (3)
(b) A geometric sequence has first term 81 and common ratio $\frac{2}{3}$ . Work out the first term of the sequence that is not a whole number. (2)

Answers and marks

(a) $21$

(b) Add 4

(c) No: the terms are 1 more than a multiple of 4, and 50 is not.

M1 Continuing the sequence past 50 (49 and 53), or noticing each term is 1 more than a multiple of 4.
C1 "No", with 49 and 53 shown (or 50 not 1 more than a multiple of 4).

(a) $45$

(b) $0$

(a) $T_n + T_{n+1} = (n + 1)^2$

(b) $\frac{32}{3}$

Term rules, special and geometric sequences

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