Answer Key
Sequences
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Write down the next two terms in the sequence: $3,\ 7,\ 11,\ 15,\ \ldots$. | 19, 23 |
| 2 | Write down the next two terms in the sequence: $2,\ 6,\ 18,\ 54,\ \ldots$. | 162, 486 |
| 3 | State whether the sequence $4,\ 8,\ 12,\ 16,\ \ldots$ is arithmetic, geometric, or neither. | Arithmetic ($d = 4$) |
| 4 | For the sequence $15,\ 11,\ 7,\ 3,\ \ldots$, state the common difference $d$. | $d = -4$ |
| 5 | For the sequence $64,\ 32,\ 16,\ 8,\ \ldots$, state the common ratio $r$. | $r = \dfrac{1}{2}$ |
| 6 | A sequence begins $7,\ 10,\ 13,\ 16,\ \ldots$. Write down (a) $u_1$ and (b) $u_3$. | (a) $u_1 = 7$ (b) $u_3 = 13$ |
| 7 | A sequence has first term $u_1 = 5$ and term-to-term rule "add $3$". Write down the first four terms. | 5, 8, 11, 14 |
| 8 | A sequence has $u_1 = 2$ and term-to-term rule "multiply by $3$". Write down the first four terms. | 2, 6, 18, 54 |
| 9 | The $n$th term of a sequence is $u_n = 3n + 2$. Find $u_1$ and $u_5$. | $u_1 = 5$, $u_5 = 17$ |
| 10 | Find the 6th term of the sequence $2,\ 5,\ 8,\ 11,\ \ldots$. | 17 |
| 11 | Find the $n$th term formula for the arithmetic sequence $5,\ 8,\ 11,\ 14,\ \ldots$. | $u_n = 3n + 2$ |
| 12 | Find the $n$th term formula for the sequence $20,\ 17,\ 14,\ 11,\ \ldots$. | $u_n = 23 - 3n$ |
| 13 | For the arithmetic sequence with $u_1 = 7$ and $d = 4$, find $u_{20}$. | $u_{20} = 83$ |
| 14 | For the sequence with $n$th term $u_n = 4n + 1$, find which term equals $81$. | $n = 20$ (the 20th term) |
| 15 | A geometric sequence has $u_1 = 3$ and $r = 2$. Find $u_5$. | $u_5 = 48$ |
| 16 | A sequence is defined recursively by $u_1 = 4$ and $u_{{n+1}} = u_n + 5$. Find the first four terms. | 4, 9, 14, 19 |
| 17 | A sequence is defined by $u_1 = 5$ and $u_{{n+1}} = 2 u_n$. Find $u_4$. | $u_4 = 40$ |
| 18 | A pattern uses $4$ matchsticks for shape 1, and each new shape adds $3$ more. How many matchsticks for shape 10? | 31 |
| 19 | An arithmetic sequence has $u_3 = 11$ and $u_8 = 31$. Find the common difference. | $d = 4$ |
| 20 | Find the $n$th term formula for the geometric sequence $4,\ 12,\ 36,\ 108,\ \ldots$. | $u_n = 4 \cdot 3^{n-1}$ |
| 21 | An arithmetic sequence has $u_5 = 17$ and $u_{15} = 47$ where $n = 15$. Find $u_1$ and $d$. | $u_1 = 5$, $d = 3$ |
| 22 | The $n$th term of a sequence is $u_n = 7n + 3$. Find the first term greater than $100$. | $u_{14} = 101$ |
| 23 | A sequence is defined by $u_1 = 5$ and $u_{{n+1}} = 2u_n - 3$. Find $u_4$. | $u_4 = 19$ |
| 24 | A geometric sequence has $u_2 = 6$ and $u_4 = 54$ (both positive). Find $u_1$ and $r$. | $u_1 = 2$, $r = 3$ |
| 25 | A sequence is given by $1,\ 4,\ 9,\ 16,\ 25,\ \ldots$. Find the next term and describe the pattern. | 36; square numbers $u_n = n^2$ |
| 26 | A sequence is defined by $u_1 = 1$, $u_2 = 2$ and $u_{{n+1}} = u_n + u_{{n-1}}$. Find $u_6$. | $u_6 = 13$ |
| 27 | Which term of the sequence $u_n = 8n - 5$ equals $99$? | $n = 13$ |
| 28 | The first three terms of a sequence are $7,\ 4,\ 1,\ -2,\ \ldots$. (a) State the type. (b) Find $u_n$. | (a) Arithmetic, $d = -3$. (b) $u_n = 10 - 3n$ |
| 29 | Find the sum of the first $10$ terms of the arithmetic sequence with $u_1 = 3$ and $d = 5$. | $S_{10} = 255$ |
| 30 | A geometric sequence has $u_1 = 16$ and $r = \dfrac{1}{2}$. Find $u_4$. | $u_4 = 2$ |
| 31 | An arithmetic sequence has $u_4 = 14$ and $u_{14} = 44$ where $n = 14$. Find $u_n$ as a formula. | $u_n = 3n + 2$ |
| 32 | Find the sum of all multiples of $7$ between 1 and $100$ inclusive. | 735 |
| 33 | A geometric sequence has $u_3 = 12$ and $u_6 = 96$ (both positive). Find $u_1$ and $r$. | $r = 2$, $u_1 = 3$ |
| 34 | Sequence A has $u_n = 5n + 2$ and sequence B has $u_n = 8n - 7$. For which value of $n$ do they have the same term? | $n = 3$ (both equal 17) |
| 35 | A sequence is defined by $u_1 = 8$ and $u_{{n+1}} = \dfrac{u_n}{2} + 3$. Find $u_4$. | $u_4 = 6.25$ (i.e. $\dfrac{25}{4}$) |
| 36 | Find the sum of the first $6$ terms of the geometric sequence with $u_1 = 2$ and $r = 3$. | $S_6 = 728$ |
| 37 | The first four terms of a sequence are $2,\ 6,\ 12,\ 20,\ \ldots$. Find the next term and a quadratic formula for $u_n$. | 30; $u_n = n^2 + n$ |
| 38 | A bacterial colony of $50$ cells doubles every hour. (a) Write the number of cells $u_n$ after $n$ hours. (b) After how many full hours will the population exceed $1000$? | (a) $u_n = 50 \cdot 2^n$. (b) After 5 hours ($u_5 = 1600$) |
| 39 | Maya saves £$5$ in week 1 and £$3$ more each week than the previous week. In which week does her **weekly** saving first exceed £$50$? | Week 17 (weekly = £53) |
| 40 | A sequence is defined by $u_1 = 1$ and $u_{{n+1}} = u_n + 2n + 1$. Find $u_5$. | $u_5 = 25$ |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Write down the next two terms in the sequence: $5,\ 9,\ 13,\ 17,\ \ldots$. | 21, 25 |
| 2 | Write down the next two terms in the sequence: $3,\ 12,\ 48,\ 192,\ \ldots$. | 768, 3072 |
| 3 | State whether the sequence $1,\ 2,\ 4,\ 8,\ \ldots$ is arithmetic, geometric, or neither. | Geometric ($r = 2$) |
| 4 | For the sequence $20,\ 14,\ 8,\ 2,\ \ldots$, state the common difference $d$. | $d = -6$ |
| 5 | For the sequence $81,\ 27,\ 9,\ 3,\ \ldots$, state the common ratio $r$. | $r = \dfrac{1}{3}$ |
| 6 | A sequence begins $12,\ 18,\ 24,\ 30,\ \ldots$. Write down (a) $u_1$ and (b) $u_3$. | (a) $u_1 = 12$ (b) $u_3 = 24$ |
| 7 | A sequence has first term $u_1 = 2$ and term-to-term rule "add $7$". Write down the first four terms. | 2, 9, 16, 23 |
| 8 | A sequence has $u_1 = 5$ and term-to-term rule "multiply by $2$". Write down the first four terms. | 5, 10, 20, 40 |
| 9 | The $n$th term of a sequence is $u_n = 4n + 1$. Find $u_1$ and $u_5$. | $u_1 = 5$, $u_5 = 21$ |
| 10 | Find the 6th term of the sequence $7,\ 12,\ 17,\ 22,\ \ldots$. | 32 |
| 11 | Find the $n$th term formula for the arithmetic sequence $3,\ 9,\ 15,\ 21,\ \ldots$. | $u_n = 6n - 3$ |
| 12 | Find the $n$th term formula for the sequence $50,\ 45,\ 40,\ 35,\ \ldots$. | $u_n = 55 - 5n$ |
| 13 | For the arithmetic sequence with $u_1 = 12$ and $d = 3$, find $u_{25}$. | $u_{25} = 84$ |
| 14 | For the sequence with $n$th term $u_n = 5n + 2$, find which term equals $102$. | $n = 20$ (the 20th term) |
| 15 | A geometric sequence has $u_1 = 2$ and $r = 3$. Find $u_5$. | $u_5 = 162$ |
| 16 | A sequence is defined recursively by $u_1 = 1$ and $u_{{n+1}} = u_n + 6$. Find the first four terms. | 1, 7, 13, 19 |
| 17 | A sequence is defined by $u_1 = 4$ and $u_{{n+1}} = 3 u_n$. Find $u_4$. | $u_4 = 108$ |
| 18 | A pattern uses $5$ matchsticks for shape 1, and each new shape adds $4$ more. How many matchsticks for shape 10? | 41 |
| 19 | An arithmetic sequence has $u_3 = 9$ and $u_8 = 24$. Find the common difference. | $d = 3$ |
| 20 | Find the $n$th term formula for the geometric sequence $5,\ 10,\ 20,\ 40,\ \ldots$. | $u_n = 5 \cdot 2^{n-1}$ |
| 21 | An arithmetic sequence has $u_5 = 14$ and $u_{15} = 44$ where $n = 15$. Find $u_1$ and $d$. | $u_1 = 2$, $d = 3$ |
| 22 | The $n$th term of a sequence is $u_n = 5n + 2$. Find the first term greater than $200$. | $u_{40} = 202$ |
| 23 | A sequence is defined by $u_1 = 4$ and $u_{{n+1}} = 2u_n - 2$. Find $u_4$. | $u_4 = 18$ |
| 24 | A geometric sequence has $u_2 = 8$ and $u_4 = 32$ (both positive). Find $u_1$ and $r$. | $u_1 = 4$, $r = 2$ |
| 25 | A sequence is given by $1,\ 8,\ 27,\ 64,\ \ldots$. Find the next term and describe the pattern. | 125; cube numbers $u_n = n^3$ |
| 26 | A sequence is defined by $u_1 = 2$, $u_2 = 3$ and $u_{{n+1}} = u_n + u_{{n-1}}$. Find $u_6$. | $u_6 = 21$ |
| 27 | Which term of the sequence $u_n = 6n - 1$ equals $89$? | $n = 15$ |
| 28 | The first three terms of a sequence are $10,\ 6,\ 2,\ -2,\ \ldots$. (a) State the type. (b) Find $u_n$. | (a) Arithmetic, $d = -4$. (b) $u_n = 14 - 4n$ |
| 29 | Find the sum of the first $12$ terms of the arithmetic sequence with $u_1 = 2$ and $d = 3$. | $S_{12} = 222$ |
| 30 | A geometric sequence has $u_1 = 81$ and $r = \dfrac{1}{3}$. Find $u_4$. | $u_4 = 3$ |
| 31 | An arithmetic sequence has $u_4 = 11$ and $u_{14} = 41$ where $n = 14$. Find $u_n$ as a formula. | $u_n = 3n - 1$ |
| 32 | Find the sum of all multiples of $5$ between 1 and $200$ inclusive. | 4100 |
| 33 | A geometric sequence has $u_3 = 18$ and $u_6 = 486$ (both positive). Find $u_1$ and $r$. | $r = 3$, $u_1 = 2$ |
| 34 | Sequence A has $u_n = 5n + 2$ and sequence B has $u_n = 7n - 6$. For which value of $n$ do they have the same term? | $n = 4$ (both equal 22) |
| 35 | A sequence is defined by $u_1 = 12$ and $u_{{n+1}} = \dfrac{u_n}{2} + 2$. Find $u_4$. | $u_4 = 5$ |
| 36 | Find the sum of the first $5$ terms of the geometric sequence with $u_1 = 3$ and $r = 2$. | $S_5 = 93$ |
| 37 | The first four terms of a sequence are $3,\ 8,\ 15,\ 24,\ \ldots$. Find the next term and a quadratic formula for $u_n$. | 35; $u_n = n^2 + 2n$ |
| 38 | A bacterial colony of $100$ cells doubles every hour. (a) Write the number of cells $u_n$ after $n$ hours. (b) After how many full hours will the population exceed $5000$? | (a) $u_n = 100 \cdot 2^n$. (b) After 6 hours ($u_6 = 6400$) |
| 39 | Maya saves £$4$ in week 1 and £$2$ more each week than the previous week. In which week does her **weekly** saving first exceed £$40$? | Week 20 (weekly = £42) |
| 40 | A sequence is defined by $u_1 = 2$ and $u_{{n+1}} = u_n + 2n + 1$. Find $u_5$. | $u_5 = 26$ |
Problems — Worked Solutions
**Theatre seating.** A theatre has 20 seats in row 1, 23 seats in row 2, 26 seats in row 3, and so on, each row adding 3 seats. (a) How many seats are in row 12? (b) Which row is the first to have at least 50 seats? (c) How many seats are there in total in the first 12 rows?
(a) 53 (b) Row 11 (51 seats) (c) 438
**Bouncing ball.** A ball is dropped from a height of 80 cm. Each bounce reaches $\frac{3}{4}$ of the previous bounce height. (a) Write the height of the $n$th bounce as a geometric sequence. (b) What is the height of the 5th bounce, to the nearest mm? (c) After how many bounces is the height less than 10 cm for the first time?
(a) $h_n = 80 \cdot (\tfrac{3}{4})^n$ (b) 19.0 cm (190 mm) (c) 8th bounce (8.0 cm)
**Salary growth.** A graduate has two job offers. - **Company A:** starting salary £24,000, with annual increases of £1,500. - **Company B:** starting salary £22,000, with annual increases of 5% on the previous year's salary. (a) Write a formula for the salary $A_n$ at company A and $B_n$ at company B in year $n$. (b) In which year does B first overtake A? (c) What is the total earned at each company over the first 5 years?
(a) $A_n = 22500 + 1500n$; $B_n = 22000 \cdot (1.05)^{n-1}$ (b) Year 17 (c) A: £135,000; B: £121,550
**Shifted pattern.** The diagram below shows the first three patterns made from dots. Pattern 1: 4 dots (square corners) Pattern 2: 7 dots Pattern 3: 10 dots (a) How many dots are in pattern $n$? (b) Pattern $k$ has 88 dots. Find $k$. (c) A student says "Pattern 100 has 304 dots." Is she correct? Show your reasoning.
(a) $u_n = 3n + 1$ (b) $k = 29$ (c) No — pattern 100 has 301 dots
**Geometric shrinkage.** A piece of paper has area 800 cm². It is folded in half repeatedly so that the new exposed top area halves each time. (a) Write a sequence for the visible area $A_n$ after $n$ folds. (b) After how many folds is the visible area less than 1 cm²? (c) Could the area ever be exactly 0 cm²? Justify mathematically.
(a) $A_n = 800 \cdot (\tfrac{1}{2})^n$ (b) 10 folds (c) No — only approaches 0 in the limit
**Mixed arithmetic/geometric.** A sequence has $u_1 = 6$. The differences between consecutive terms form a geometric sequence: $u_2 - u_1 = 4$, $u_3 - u_2 = 8$, $u_4 - u_3 = 16$, and so on. (a) Find $u_5$. (b) Find a closed form for $u_n$.
(a) $u_5 = 66$ (b) $u_n = 6 + 4(2^{n-1} - 1) = 2 + 2^{n+1}$
**Card stacking.** Sienna builds a house of cards. Level 1 (top) uses 2 cards. Level 2 needs 5 cards. Level 3 needs 8 cards. Each level adds 3 more cards than the level above. (a) How many cards are needed for level $n$? (b) How many cards are needed to build all levels from 1 to 10? (c) Sienna has 200 cards. Including all levels, what is the largest house she can build?
(a) $u_n = 3n - 1$ (b) 155 cards (c) 11 levels (uses 187 cards)
**Identifying the type.** For each sequence, state whether it is arithmetic, geometric, or neither, and find a formula or rule for $u_n$. (a) $5, 9, 13, 17, 21, \ldots$ (b) $3, 6, 12, 24, 48, \ldots$ (c) $1, 4, 9, 16, 25, \ldots$ (d) $2, 5, 11, 23, 47, \ldots$
(a) Arithmetic, $u_n = 4n+1$ (b) Geometric, $u_n = 3 \cdot 2^{n-1}$ (c) Neither (square), $u_n = n^2$ (d) Neither, recursive $u_{n+1} = 2u_n + 1$
**Two sequences meeting.** Sequence A: $u_n = 4n + 3$. Sequence B: starts at 47 and decreases by 2 each term. (a) Write a formula for sequence B. (b) Find the value of $n$ for which $A_n = B_n$. (c) What is the value of the common term?
(a) $B_n = 49 - 2n$ (b) $n = 7\tfrac{2}{3}$ — no integer solution; see working (c) No integer common term
**Recursive notation challenge.** A sequence is defined by $$u_1 = 1, \quad u_2 = 3, \quad u_{n+1} = u_n + 2 u_{n-1}.$$ (a) Find $u_3$, $u_4$, $u_5$, $u_6$. (b) Show that all terms are odd integers. (c) Calculate $\dfrac{u_{n+1}}{u_n}$ for $n = 1, 2, 3, 4, 5$. What do you notice?
(a) 5, 11, 21, 43 (b) See working (c) Ratios: 3, 1.67, 2.2, 1.91, 2.05 — approach 2
**Sum of squares puzzle.** Consider the sequence $1, 4, 9, 16, 25, \ldots$ of square numbers. (a) Sum the first five terms. (b) The formula $\displaystyle S_n = \frac{n(n+1)(2n+1)}{6}$ gives the sum of the first $n$ square numbers. Verify this for $n = 5$. (c) Find the sum of the first 20 square numbers. (d) Find the sum: $4 + 9 + 16 + 25 + \ldots + 400$.
(a) 55 (b) 55 ✓ (c) 2870 (d) 2869
**Aesthetics of growth.** A nautilus shell grows in a logarithmic spiral. Each chamber is $\phi$ times the previous, where $\phi = \frac{1+\sqrt{5}}{2} \approx 1.618$ (the golden ratio). The smallest chamber has area $1 \text{ mm}^2$. (a) Write a formula for the area $A_n$ of the $n$th chamber. (b) Estimate the area of the 8th chamber (3 s.f.). (c) The total area of the first $n$ chambers is given by a geometric sum. Find the total area of the first 8 chambers (3 s.f.).
(a) $A_n = \phi^{n-1}$ (b) $\approx 29.0$ mm² (c) $\approx 74.4$ mm²