19. Sequences (Linear) - Cardiff Tutor Company

Question 1

LEVEL 4

Find the $n^{th}$ term given the following sequence

$-1,-10,-19,-28,-37\dots$

Select the correct answer from the list below.

A: $-9n+8$

B: $9n+8$

C: $-9n-8$

D: $-8n+9$

CORRECT ANSWER: A: $-9n+8$

WORKED SOLUTION:

$n=1$ , then $-9\times1=-9$ so $-1-(-9)=-1+9=8$

$n=2$ , then $-9\times2=-18$ so $-10-(-18)=-10+18=8$

$n=3$ , then $-9\times3=-27$ so $-19-(-27)=-19+27=8$

$n=4$ , then $-9\times4=-36$ so $-28-(-36)=-28+36=8$

A pattern emerges whereby to obtain the nth-term of the sequence you can use the formula: $-9n+8$

Question 2

LEVEL 4

Give the first $5$ numbers in the sequence with formula $17-6n$

Select the correct answer from the list below.

A: $11,5,-1,-7,-13$

B: $-11,-5,1,7,13$

C: $11,6,0,-6,-12$

D: $17, 11, 5, -1, -7$

CORRECT ANSWER: A: $11,5,-1,-7,-13$

WORKED SOLUTION:

$n=1$ , then $17-(6\times1)=17-6=11$

$n=2$ , then $17-(6\times2)=17-12=5$

$n=3$ , then $17-(6\times3)=17-18=-1$

$n=4$ , then $17-(6\times4)=17-24=-7$

$n=5$ , then $17-(6\times5)=17-30=-13$

Hence the first $5$ numbers in the sequence with formula $17-6n$ are $11,5,-1,-7,-13$

Question 3

LEVEL 4

A sequence reads $12,14,16,18,20\dots$

State the $n^{th}$ -term and find if $174$ is a term in the sequence.

Select the correct answer from the list below.

A: Rule $= 2n+10 , \,\, 174$ is not a term in the sequence

B: Rule $= 2n+10 , \,\, 174$ is a term in the sequence

C: Rule $= 2n+12 , \,\, 174$ is not a term in the sequence

D: Rule $= 2n+12 , \,\, 174$ is a term in the sequence

CORRECT ANSWER: B: Rule $= 2n+10 , \,\, 174$ is a term in the sequence

WORKED SOLUTION:

Use $n=1, n=2,n=3$ etc to find how much you need to add/subtract to $5n$ to get to the sequence.

$n=1$ , then $2\times1=2$ so $12-2=10$

$n=2$ , then $2\times2=4$ so $14-4=10$

$n=3$ , then $2\times3=6$ so $16-6=10$

If $n=4$ , then $2\times4=8$ so $18-8=10$

A pattern emerges whereby to obtain the nth-term of the sequence you can use the formula: $2n+10$

Now equate formula to term in question $2n+10=174$ and solve

Subtract $10$ on both sides $2n=174-10=164$

Divide both sides by $2$ , hence $n=\dfrac{164}{2}=82$

If $n=\,$ integer then the term in the question is part of the sequence, if $n \neq$ integer then the term in the question is not part of the sequence.

$82$ is an integer, hence the number $174$ is a term in the sequence $2n+10$ .

Question 4

LEVEL 4

Give the first $5$ numbers in the sequence with the formula $2n+7$

Select the correct answer from the list below.

A: $9,11,13,15,17$

B: $2,4,6,8,10$

C: $7,9,11,13,15$

D: $2,3,4,5,6$

CORRECT ANSWER: A: $9,11,13,15,17$

WORKED SOLUTION:

n=1

, then

(2\times1)+7=9

$n=2$ , then $(2\times2)+7=11$

$n=3$ , then $(2\times3)+7=13$

$n=4$ , then $(2\times4)+7=15$

$n=5$ , then $(2\times5)+7=17$

Answer $= 9,11,13,15,17$

Question 5

LEVEL 4

Find the $n^{th}$ term given the following sequence $2,9,16,23,30\dots$

Select the correct answer from the list below.

A: $-5n+7$

B: $5-7n$

C: $5n-7$

D: $7n-5$

CORRECT ANSWER: D: $7n-5$

WORKED SOLUTION:

$n=1$ , then $7\times1=7$ so $2-7=-5$

$n=2$ , then $7\times2=14$ so $9-14=-5$

$n=3$ , then $7\times3=21$ so $16-21=-5$

A pattern emerges whereby to obtain the $n^{th}$ term of the sequence you can use the formula: $7n-5$

Question 6

LEVEL 4

A sequence reads $12,25,38,51, 64\dots$

State the $n^{th}$ -term and find if $135$ is a term in the sequence.

Select the correct answer from the list below.

A: Rule $= 13n-1, \,\, 135$ is not a term in the sequence

B: Rule $= 13n-1, \,\, 135$ is a term in the sequence

C: Rule $= 13n+1, \,\, 135$ is not a term in the sequence

D: Rule $= 13n-1, \,\, 135$ is a term in the sequence

CORRECT ANSWER: A: Rule $= 13n-1, \,\, 135$ is not a term in the sequence

WORKED SOLUTION:

$n=1$ , then $13\times1=13$ so $12-13=-1$

$n=2$ , then $13\times2=26$ so $25-26=-1$

$n=3$ , then $13\times3=39$ so $38-39=-1$

$n=4$ , then $13\times4=52$ so $51-52=-1$

A pattern emerges whereby to obtain the nth-term of the sequence you can use the formula: $13n-1$

Now equate formula to term in question $13n-1=135$ and solve

Subtract $1$ on both sides $13n=135-1=134$

Divide both sides by $13$ , hence $n=\dfrac{134}{13}=10.31$ (to $2$ dp)

If $n=\,$ integer then the term in the question is part of the sequence, if $n \neq$ integer then the term in the question is not part of the sequence.

$10.31$ is not an integer hence number $135$ is not a term in the sequence $13n-1$

Question 7

LEVEL 4

A sequence reads $-5,-16,-27,-38,-49\dots$

State the $n^{th}$ -term and find if $-159$ is a term in the sequence.

Select the correct answer from the options below:

A: Rule $= 6-11n, \,\, -159$ is not a term in the sequence

B: Rule $= 6-11n, \,\, -159$ is a term in the sequence

C: Rule $= -6-11n, \,\, -159$ is not a term in the sequence

D: Rule $= -6-11n, \,\, -159$ is a term in the sequence

CORRECT ANSWER: B: Rule $= 6-11n, \,\, -159$ is a term in the sequence

WORKED SOLUTION:

$n=1$ , then $-11\times1=-11$ so – $5-(-11)=-5+11=6$

$n=2$ , then $-11\times2=-22$ so $-16-(-22)=-16+22=6$

$n=3$ , then $-11\times3=-33$ so $-27-(-33)=-27+33=6$

$n=4$ , then $-11\times4=-44$ so $-38-(-44)=-38+44=6$

A pattern emerges whereby to obtain the $n^{th}$ -term of the sequence you can use the formula: $-11n+6=6-11n$

Now equate formula to term in question $6-11n=-159$ and solve

Add $11n$ on both sides $6=11n-159$

Add $159$ on both sides $6+159=11n$

$11n=159+6=165$ hence $n=\dfrac{165}{11}=15$

If $n=\,$ integer then the term in the question is part of the sequence, if $n \neq$ integer then the term in the question is not part of the sequence.

$15$ is an integer hence number $-159$ is a term in the sequence $6-11n$ (note: it’s the $15$ th term)

Question 8

LEVEL 4

A geometric sequence reads $1, 3, 9, 27,\dots$

Work out the next value in the sequence.

Select the correct answer from the options below:

A: $36$

B: $81$

C: $243$

D: $54$

CORRECT ANSWER: B: $81$

WORKED SOLUTION:

Each term in the sequence is multiplied by a common ratio of $3$ .

Therefore to find the next term in the sequence, we multiply the previous term by $3$ .

So $27 \times 3 = 81$ is our next term.

Question 9

LEVEL 4

A geometric sequence reads $\sqrt{2}, 2 , 2\sqrt{2}, 4,…$

Work out the next value in the sequence.

Select the correct answer from the options below:

A: $6$

B: $8$

C: $4\sqrt{2}$

D: $4 + \sqrt{2}$

CORRECT ANSWER: C: $4\sqrt{2}$

WORKED SOLUTION:

Each term in the sequence is multiplied by a common ratio of $\sqrt{2}$ .

Therefore to find the next term in the sequence, we multiply the previous term by $\sqrt{2}$ .

So $4 \times \sqrt{2} = 4\sqrt{2}$ is our next term.

Question 10

LEVEL 4

Identify which of the following sequences is geometric.

Select the correct answer from the options below:

A: $2, 4, 6, 8, 10, ...$

B: $2, 4, 8, 12, 20, ...$

C: $2, 4, 8, 16, 24, ...$

D: $2, 4, 8, 16, 32, ...$

CORRECT ANSWER: D: $2, 4, 8, 16, 32, ...$

WORKED SOLUTION:

Each term in the sequence is multiplied by a common ratio of $2$ , so it is geometric.

Question 11

LEVEL 4

Identify which of the following sequences is the sequence of cubic numbers.

Select the correct answer from the options below:

A: $1,3,6,10,15, ...$

B: $1,8,27,64,125, ...$

C: $1,1,2,3,5, ...$

D: $1,4,9,16,25, ...$

CORRECT ANSWER: B: $1,8,27,64,125, ...$

WORKED SOLUTION:

Each term in the sequence is a cubic number.

Question 12

LEVEL 4

Identify which of the following sequences is the sequence of Fibonacci numbers.

Select the correct answer from the options below:

A: $1,3,6,10,15, ...$

B: $1,8,27,64,125, ...$

C: $1,1,2,3,5, ...$

D: $1,4,9,16,25, ...$

CORRECT ANSWER: C: $1,1,2,3,5, ...$

WORKED SOLUTION:

Each term in the sequence is the sum of the $2$ previous terms.

Question 13

LEVEL 4

Identify which of the following sequences is the sequence of triangular numbers.

Select the correct answer from the options below:

A: $1,3,6,10,15, ...$

B: $1,8,27,64,125, ...$

C: $1,1,2,3,5, ...$

D: $1,4,9,16,25, ...$

CORRECT ANSWER: A: $1,3,6,10,15, ...$

WORKED SOLUTION:

Each term in the sequence is a triangular number, a number that can be represented as an equilateral triangle of dots.