Teach Me Maths

nth Term of Linear Sequences

Question 11 mark

The linear sequence begins 4,7,10,13,\ldots. Find the next term.

Hint

Compare two neighbouring terms, then check that the same change repeats.

Worked solution

The linear sequence begins 4,7,10,13,\ldots. Find the next term.

  1. Each term is 3 more than the previous one.
  2. Add 3 to the last given term: 13+3=16.

Question 21 mark

Find the common difference in the linear sequence 18,14,10,6,\ldots.

Hint

Use next term minus previous term; a decrease gives a negative difference.

Worked solution

Find the common difference in the linear sequence 18,14,10,6,\ldots.

  1. Subtract a term from the next one: 14-18=-4.
  2. The terms decrease by 4 each time, so the common difference is -4.

Question 31 mark

Find the missing term in the linear sequence 6,11,\square,21,\ldots.

Hint

A linear sequence has the same change at every step.

Worked solution

Find the missing term in the linear sequence 6,11,\square,21,\ldots.

  1. The first increase is 11-6=5.
  2. Add 5 again: 11+5=16. Check: 16+5=21.

Question 41 mark

The nth term is 3n+4. Find the first term.

Hint

The first position uses n = 1.

Worked solution

The nth term is 3n+4. Find the first term.

  1. The first term has n=1.
  2. Substitute: 3\times1+4=7.

Question 51 mark

The nth term is 5n-2. Find the fourth term.

Hint

Replace n with the position requested, not with the first term.

Worked solution

The nth term is 5n-2. Find the fourth term.

  1. Use n=4: 5\times4-2=20-2=18.
  2. The fourth term is 18.

Question 62 marks

Here are the first four terms of an arithmetic sequence: 2,6,10,14,\ldots

Find an expression, in terms of n, for the nth term.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Compare the sequence with the multiples of its common difference.

Worked solution

Here are the first four terms of an arithmetic sequence: 2,6,10,14,\ldots

Find an expression, in terms of n, for the nth term.

  1. The common difference is 4. The multiples 4n start 4,8,12,16.
  2. Every term is 2 less, so the nth term is 4n-2.
  3. Check: when n=1, 4n-2=2; when n=4, it is 14.

Question 72 marks

Here are the first four terms of an arithmetic sequence: 8,11,14,17,\ldots

Find an expression, in terms of n, for the nth term.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Compare the first term with the first multiple of the common difference.

Worked solution

Here are the first four terms of an arithmetic sequence: 8,11,14,17,\ldots

Find an expression, in terms of n, for the nth term.

  1. The common difference is 3, so start with 3n.
  2. Its first term is 3, and the required first term is 8. Add 8-3=5.
  3. The nth term is 3n+5. Check: 3\times4+5=17.

Question 82 marks

Find an expression, in terms of n, for the nth term of 5,10,15,20,\ldots.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Compare the terms with a multiplication table.

Worked solution

Find an expression, in terms of n, for the nth term of 5,10,15,20,\ldots.

  1. The difference is 5. The multiples 5n begin 5,10,15,20.
  2. They already match, so no extra adjustment is needed. The rule is 5n.

Question 92 marks

Find an expression, in terms of n, for the nth term of the arithmetic sequence 9,16,23,30,\ldots.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Find the common difference first, then compare the sequence with the multiples of that number.

Worked solution

Find an expression, in terms of n, for the nth term of the arithmetic sequence 9,16,23,30,\ldots.

  1. The common difference is 7, so start with 7n.
  2. At n=1, 7n gives 7. Add 2 to get the first term 9.
  3. The rule is 7n+2. Check the fourth term: 7\times4+2=30.

Question 101 mark

The nth term is 2n+7. Find the 10th term.

Hint

The nth-term rule lets you jump straight to the required position.

Worked solution

The nth term is 2n+7. Find the 10th term.

  1. Substitute n=10: 2\times10+7=27.
  2. The 10th term is 27.

Question 113 marks

The table shows the first four terms of an arithmetic sequence.

(a)

Find the nth term.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Find the 20th term.

1 mark

Hint

For (a), find the change between adjacent terms and compare with its multiples. For (b), use the term position in your rule.

Worked solution

The table shows the first four terms of an arithmetic sequence.

Part (a)

Find the nth term.

  1. Start with 4n. Its first term is 4, so add 9-4=5.
  2. The rule is 4n+5.

Part (b)

Find the 20th term.

  1. Use the rule from part (a): 4\times20+5=80+5=85.
  2. The 20th term is 85.

Question 122 marks

Find an expression, in terms of n, for the nth term of the arithmetic sequence -3,2,7,12,\ldots.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

A sequence can increase even if its first term is negative.

Worked solution

Find an expression, in terms of n, for the nth term of the arithmetic sequence -3,2,7,12,\ldots.

  1. The difference is 5, so start with 5n.
  2. At n=1, the value 5 must become -3: subtract 8.
  3. The rule is 5n-8. Check: 5\times4-8=12.

Question 133 marks

Here are the first four terms of an arithmetic sequence: 7,12,17,22,\ldots

Which term of the sequence is 62?

Hint

First find the nth term, then set it equal to the required value.

Worked solution

Here are the first four terms of an arithmetic sequence: 7,12,17,22,\ldots

Which term of the sequence is 62?

  1. The common difference is 5; comparing the first term gives the rule 5n+2.
  2. Set it equal to 62: 5n+2=62, so 5n=60 and n=12.
  3. 62 is the 12th term.

Question 142 marks

The nth term of a sequence is 4n-1.

Is 50 a term of the sequence? Choose the statement with a valid reason.

Hint

Work out the exact position that 50 would need; do not round it.

Worked solution

The nth term of a sequence is 4n-1.

Is 50 a term of the sequence? Choose the statement with a valid reason.

  1. Yes, because 50 is larger than the first term.
  2. Yes, because 12.75 rounds to 13.
  3. No, because solving for the position gives n = 12.75.
  4. No, because all the terms are multiples of 4.
  1. Set 4n-1=50. Add 1 and divide by 4: n=51\div4=12.75.
  2. A term position must be a positive whole number, so 50 is not a term.

Question 152 marks

Here are the first three dot patterns in a sequence. The patterns continue in the same way.

Work out the number of dots in pattern 12.

Hint

Separate the dots repeated in every new group from the dot that stays fixed.

Worked solution

Here are the first three dot patterns in a sequence. The patterns continue in the same way.

Work out the number of dots in pattern 12.

  1. The growing groups contribute 5n dots. The fixed dot adds 1 once.
  2. The rule is 5n+1. At n=12, 5\times12+1=61.
  3. Pattern 12 has 61 dots.

Question 16Challenge3 marks

A student says the nth term of 6,10,14,18,\ldots is 4n+6.

(a)

Which check shows that the proposed rule is wrong?

1 mark

(b)

Write the correct nth-term rule.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

For (a), test the first position. For (b), compare the actual sequence with multiples of its common difference.

Worked solution

A student says the nth term of 6,10,14,18,\ldots is 4n+6.

Part (a)

Which check shows that the proposed rule is wrong?

  1. At n = 1, the rule gives 10 instead of 6.
  2. The difference is 6, not 4.
  3. An nth-term rule cannot contain addition.
  4. The rule has to give 0 when n = 0.
  1. Substitute n=1 into the proposed rule: 4\times1+6=10.
  2. This should give the first term, but the required first term is 6.

Part (b)

Write the correct nth-term rule.

  1. The multiples 4n begin 4,8,12,16.
  2. Add 2 to match 6,10,14,18, so the rule is 4n+2.

Question 17Challenge4 marks

The nth term of a sequence is 5n+3.

(a)

At which position does 98 occur?

2 marks

(b)

Is 100 a term? Choose the statement with a valid reason.

2 marks

Hint

Solve for the position separately for each number; being close to a term does not make a number a term.

Worked solution

The nth term of a sequence is 5n+3.

Part (a)

At which position does 98 occur?

  1. Solve 5n+3=98: subtract 3 to get 5n=95, then divide by 5.
  2. n=19, so 98 is the 19th term.

Part (b)

Is 100 a term? Choose the statement with a valid reason.

  1. Yes, because it is only 2 more than 98.
  2. No, because the required position is 19.4.
  3. Yes, because 19.4 rounds to 19.
  4. No, because 100 is even.
  1. Solve 5n+3=100: 5n=97, so n=19.4.
  2. This is not a whole-number position, so 100 is not a term.

Question 18Challenge4 marks

Here are the first four terms of a decreasing arithmetic sequence: 31,27,23,19,\ldots

The sequence continues in the same way.

(a)

At which position is the first negative term?

3 marks

(b)

What is the value of that first negative term?

1 mark

Hint

Find the rule, then check the positions either side of where the terms cross zero.

Worked solution

Here are the first four terms of a decreasing arithmetic sequence: 31,27,23,19,\ldots

The sequence continues in the same way.

Part (a)

At which position is the first negative term?

  1. The difference is -4, giving nth term 35-4n.
  2. The 8th term is 35-4\times8=3, which is positive. The 9th term is 35-4\times9=-1, which is negative.
  3. Because each term is 4 less than the previous one, the first negative term is at position 9.

Part (b)

What is the value of that first negative term?

  1. Use position 9 from part (a): 35-4\times9=35-36=-1.
  2. The first negative value is -1.

Question 19Challenge3 marks

The 3rd term of an arithmetic sequence is 14. The 4th term is 19.

(a)

Find an expression, in terms of n, for the nth term.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Find the 30th term.

1 mark

Hint

The gap between two terms next to each other is the common difference. Count back to find the 1st term.

Worked solution

The 3rd term of an arithmetic sequence is 14. The 4th term is 19.

Part (a)

Find an expression, in terms of n, for the nth term.

  1. The common difference is 19-14=5, so start with 5n.
  2. Counting back: the 2nd term is 14-5=9 and the 1st term is 9-5=4.
  3. At n=1, 5n gives 5. Subtract 1 to get 4, so the nth term is 5n-1.
  4. Check: 5\times3-1=14 and 5\times4-1=19.

Part (b)

Find the 30th term.

  1. Substitute n=30 into 5n-1: 5\times30-1=149.
  2. The 30th term is 149.

Question 20Challenge4 marks

The nth term of sequence A is 4n+1. The nth term of sequence B is 6n-9.

The two sequences have equal values at the same position.

(a)

Find the position n.

3 marks

(b)

Find the common value.

1 mark

Hint

At the same position, both rules use the same n. Set the two expressions equal, then calculate the term value.

Worked solution

The nth term of sequence A is 4n+1. The nth term of sequence B is 6n-9.

The two sequences have equal values at the same position.

Part (a)

Find the position n.

  1. Set the two values equal: 4n+1=6n-9.
  2. Subtract 4n and add 9: 10=2n. Divide by 2: n=5.
  3. They have the same value at position 5.

Part (b)

Find the common value.

  1. Substitute n=5 into A: 4\times5+1=21.
  2. Check B: 6\times5-9=21. Both 5th terms are 21.

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