Sequences and the nth term

The even numbers are the 2 times table.

The 10th one is $2 \times 10 = 20$.

(a) Each new pattern adds one square to each arm. Pattern 5 has arms 5 squares long, sharing the corner square: 9 squares.

(b) The numbers go up by 2 each time: 1, 3, 5, 7, 9, 11

Odd numbers: 1, 3, 5, 7, 9, 11, 13

The 7th one is 13.

The gaps go up by one each time: $+2, +3, +4, +5$.

Next gaps are $+6$ and $+7$: $15 + 6 = 21$, $21 + 7 = 28$

(a) The terms go up by 5, so the next term is $23 + 5 = 28$.

(b) The second term is 8 and the fourth is 18.

$8 : 18$, divide both by 2: $4 : 9$

(a) $97 - 91 = 6$, so each term is 6 less than the one before.

(b) 5th term $= 73$, 6th $= 67$, 7th $= 61$. Difference $= 73 - 61 = 12$

(c) Every term is odd (odd minus 6 stays odd). 52 is even, so it cannot appear. The terms go 55, 49, missing 52.

The differences are $20 - 16 = 4$ and $16 - 13 = 3$.

If the differences keep falling by 1, next subtract 2 then 1: $13 - 2 = 11$, $11 - 1 = 10$.

Rule: the number you take away goes down by 1 each time.

(a) The terms go up by 5: $23 + 5 = 28$, $28 + 5 = 33$

(b) The terms go 3, 8, 13, 18, ... so every term ends in 3 or 8.

50 ends in 0, so it is not a term (48 and 53 are consecutive terms).

(a)(i) Subtract 5 each time: 20, 15

(ii) Keep going: 10, 5, 0, −5 are terms 8 to 11, so $-5$ is the 11th term.

(b) $4 \times 9 + 3 = 39$

Each term is 7 more than the one before, so the next term is $23 + 7 = 30$.

The nth term is $7n - 5$.

10th term: $7 \times 10 - 5 = 65$.

(a) $1 \to 2$ is $+1$, $2 \to 4$ is $+2$, so the next step could be $+3$: $4 + 3 = 7$.

(b) The gaps grow by 1: $10 + 5 = 15$, $15 + 6 = 21$, $21 + 7 = 28$, $28 + 8 = 36$.

The 8th term is $36$.

The terms go up by 3, so start with $3n$: 3, 6, 9, ...

Each term is 2 less than that, so the nth term is $3n - 2$.

Terms: 24, 32, 40, 48, 56, 64, 72, 80, 88, ...

(a) 40 is in the sequence and is in the 5 times table.

(b) No. 80 and 88 are terms, so 85 is skipped. Also, every term is even and 85 is odd.

(a) Terms: 7, 10, 13, 16, 19, 22, ... It jumps from 19 to 22, so 21 is missed.

Or: $3n + 4 = 21 \Rightarrow 3n = 17 \Rightarrow n = 5.67$, not a whole number.

(b) Doubling: 1, 2, 4, 8, 16. Or the gaps grow by 1: 1, 2, 4, 7, 11.

Pattern $n$ has $n + 2$ grey tiles in a column and $n$ white tiles beside it.

(a) Pattern 4: 6 grey tiles in a column, 4 white tiles next to the middle ones.

(b) Total tiles $= (n + 2) + n = 2n + 2$. For $n = 20$: $2 \times 20 + 2 = 42$.

(c) Grey tiles go 3, 4, 5, ... so the expression is $n + 2$.

The terms go down by 6 each time.

32, 26, 20, 14, 8, 2, −4, −10

The first two terms below zero are −4 and −10.

The terms go up by 4, so start with $4n$.

$4n$ gives 4, 8, 12, ... Each term is 3 less, so the nth term is $4n - 3$.

The terms go up by 8, so start with $8n$: 8, 16, 24, ...

Each term of the sequence is 13 less: $8 - 13 = -5$.

nth term $= 8n - 13$

(a) The terms go up by 6, so start with $6n$: 6, 12, 18, ... Each term is 1 more, so $6n + 1$.

(b) $8 - 6n = -58$

$6n = 66$, so $n = 11$.

11 is a whole number, so yes, $-58$ is the 11th term.

Each term is the sum of the two before, so the fifth term is $3a + 5a = 8a$.

$a + 2a + 3a + 5a + 8a = 19a$

$19a = 228$, so $a = 12$.

The terms go up by 6, so start with $6n$.

$6n$ gives 6, 12, 18, 24. Each term is 1 less, so the nth term is $6n - 1$.

First sequence $2n^2 - 1$: 1, 7, 17, 31, 49, 71, ... (increasing)

Second sequence $40 - n^2$: 39, 36, 31, 24, 15, 4, -9, ... (decreasing)

The only number in both is 31. After this the first keeps rising above 31 and the second keeps falling below it, so there can be no other.

(a) Add the two previous terms: $9 + 15 = 24$, then $15 + 24 = 39$

(b) $a,\ a,\ 2a$, then $a + 2a = 3a$, $2a + 3a = 5a$, $3a + 5a = 8a$

The 6th term is $8a$.

First sequence $2n^2 - 1$: 1, 7, 17, 31, 49, 71, ... (increasing)

Second sequence $40 - n^2$: 39, 36, 31, 24, 15, 4, −9, ... (decreasing)

31 appears in both. After that the first sequence keeps rising above 31 and the second keeps falling below it, so no other number can be in both.

2016: $1.05(9500 - 250) = 9712.5$

2017: $1.05(9712.5 - 250) = 9935.625$

2018: $1.05(9935.625 - 250) = 10169.90…$

About 10170 bees.

(a) Differences: 17, 27, 37, 47; second difference 10, so $5n^2$.

Sequence − $5n^2$: $-2, 0, 2, 4, 6$, which is $2n - 4$.

nth term $= 5n^2 + 2n - 4$

(b) $u_2 = 9k + k = 10k = 4$, so $k = 0.4$

$u_3 = 0.4 \times 4 + 0.4 = 2$, $u_4 = 0.4 \times 2 + 0.4 = 1.2$

Differences: 6, 8, 10; second difference 2, so start with $n^2$.

Subtract $n^2$ (1, 4, 9, 16): leaves 2, 5, 8, 11, which is $3n - 1$.

nth term $= n^2 + 3n - 1$

First differences: 6, 10, 14, 18, 22. Second difference: 4, so the sequence starts $2n^2$.

$2n^2$: 2, 8, 18, 32, ...

Sequence minus $2n^2$: 3, 3, 3, 3, ... so the nth term is $2n^2 + 3$.

$29\,600 = 24\,000a + 800 \Rightarrow a = \frac{28\,800}{24\,000} = 1.2$

2020: $1.2 \times 29\,600 + 800 = 36\,320$

2021: $1.2 \times 36\,320 + 800 = £44\,384$

First differences: 11, 17, 23, 29. Second differences: 6, so the $n^2$ term is $\frac{6}{2}n^2 = 3n^2$.

Subtract $3n^2$ (3, 12, 27, 48, 75): leaves 7, 9, 11, 13, 15, which is $2n + 5$.

nth term $= 3n^2 + 2n + 5$.

First differences: 6, 10, 14, 18, 22. Second differences: 4, so the sequence starts $2n^2$.

$2n^2$: 2, 8, 18, 32, ...

Sequence minus $2n^2$: −3, −3, −3, ... so the nth term is $2n^2 - 3$.

First differences: 1, 3, 5, 7. Second difference 2, so start with $n^2$: 1, 4, 9, 16, 25.

Sequence minus $n^2$: −2, −4, −6, −8, −10, which is $-2n$.

nth term $= n^2 - 2n$

(a) $n = 2$: $4a + 2b = -2$

$n = 4$: $16a + 4b = 12$

Double the first: $8a + 4b = -4$. Subtract: $8a = 16$, so $a = 2$, then $b = -5$.

6th term: $2(36) - 5(6) = 72 - 30 = 42$

(b) Second differences are 2, so start with $n^2$: 1, 4, 9, 16, 25.

Sequence minus $n^2$: $-1, -2, -3, -4, -5$, which is $-n$.

nth term $= n^2 - n$

First differences: 2, 4, 6, 8; second difference 2, so start with $n^2$.

$n^2$: 1, 4, 9, 16, 25

Sequence minus $n^2$: 0, −1, −2, −3, −4, which is $-n + 1$

nth term $= n^2 - n + 1$

(a) $(x+2)-(x-4) = 6$ and $(3x+1)-(x+2) = 2x-1$, so $2x - 1 = 6$, $x = 3.5$

(b) $\dfrac{y+2}{y-4} = \dfrac{3y+1}{y+2}$

$(y+2)^2 = (3y+1)(y-4) \Rightarrow y^2 + 4y + 4 = 3y^2 - 11y - 4$

$2y^2 - 15y - 8 = 0 \Rightarrow (2y+1)(y-8) = 0$, so $y = -0.5$ or $y = 8$

(a) Ratio: $10 \div \sqrt{5} = 2\sqrt{5}$

Next term: $400\sqrt{5} \times 2\sqrt{5} = 800 \times 5 = 4000$

(b) $r^2 = \frac{5\sqrt{2}}{8} \div \frac{5\sqrt{2}}{4} = \frac{1}{2}$, so $r = \frac{1}{\sqrt{2}}$ (terms positive)

$a = \frac{5\sqrt{2}}{4} \div r^3 = \frac{5\sqrt{2}}{4} \times 2\sqrt{2} = \frac{20}{4} = 5$