nth term of linear sequences Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE nth term of linear sequences questions. See exactly how to solve problems on common difference, linear sequence, decreasing sequence, nth term.

common differencelinear sequencedecreasing sequencenth termmultiplescommon difference of 1
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
Here are the first four terms of a sequence. 3, 5, 7, 9, …3,\ 5,\ 7,\ 9,\ \ldots Write down the common difference of the sequence.

Worked solution

  1. Find the gaps between consecutive terms

    5−3=2, 7−5=2, 9−7=25 - 3 = 2,\ 7 - 5 = 2,\ 9 - 7 = 2

    Subtract each term from the one after it.

  2. Check the gaps are all the same

    d=2d = 2

    Every gap is 2, so the sequence is linear and the common difference is 2.

  3. State the common difference

    22

    The terms increase by the same amount each time.

Answer
22
Question 2
1 markeasy
Here are the first four terms of a sequence. 20, 17, 14, 11, …20,\ 17,\ 14,\ 11,\ \ldots Write down the common difference of the sequence.

Worked solution

  1. Find the gaps between consecutive terms

    17−20=−3, 14−17=−3, 11−14=−317 - 20 = -3,\ 14 - 17 = -3,\ 11 - 14 = -3

    Subtract each term from the one after it.

  2. Check the gaps are all the same

    d=−3d = -3

    Every gap is -3, so the sequence is linear and the common difference is -3.

  3. State the common difference

    −3-3

    The terms are decreasing, so the common difference is negative.

Answer
−3-3
Question 3
1 markeasy
Here are the first four terms of a sequence. 2, 4, 6, 8, …2,\ 4,\ 6,\ 8,\ \ldots Work out the nnth term of the sequence.

Worked solution

  1. Find the common difference

    4−2=2, 6−4=2, 8−6=24 - 2 = 2,\ 6 - 4 = 2,\ 8 - 6 = 2

    The gap between one term and the next is the same every time, so this is a linear (arithmetic) sequence. It goes up in 2s, so the common difference is d=2d = 2.

  2. Write down the multiples of the common difference

    2n:2, 4, 6, 82n:\quad 2,\ 4,\ 6,\ 8

    Because the common difference is 2, the nth term must start with 2n. Working out 2n for n=1n = 1, 2, 3, 4 gives 2, 4, 6, 8.

  3. Write down the nth term

    nth term=2n\text{nth term} = 2n

    Putting the 2n and the constant together gives the rule 2n.

Answer
2n2n
Question 4
1 markeasy
Here are the first four terms of a sequence. 5, 10, 15, 20, …5,\ 10,\ 15,\ 20,\ \ldots Work out the nnth term of the sequence.

Worked solution

  1. Find the common difference

    10−5=5, 15−10=5, 20−15=510 - 5 = 5,\ 15 - 10 = 5,\ 20 - 15 = 5

    The gap between one term and the next is the same every time, so this is a linear (arithmetic) sequence. It goes up in 5s, so the common difference is d=5d = 5.

  2. Write down the multiples of the common difference

    5n:5, 10, 15, 205n:\quad 5,\ 10,\ 15,\ 20

    Because the common difference is 5, the nth term must start with 5n. Working out 5n for n=1n = 1, 2, 3, 4 gives 5, 10, 15, 20.

  3. Write down the nth term

    nth term=5n\text{nth term} = 5n

    Putting the 5n and the constant together gives the rule 5n.

Answer
5n5n
Question 5
2 markseasy
Here are the first four terms of a sequence. 4, 5, 6, 7, …4,\ 5,\ 6,\ 7,\ \ldots Work out the nnth term of the sequence.

Worked solution

  1. Find the common difference

    5−4=1, 6−5=1, 7−6=15 - 4 = 1,\ 6 - 5 = 1,\ 7 - 6 = 1

    The gap between one term and the next is the same every time, so this is a linear (arithmetic) sequence. It goes up in 1s, so the common difference is d=1d = 1.

  2. Write down the multiples of the common difference

    n:1, 2, 3, 4n:\quad 1,\ 2,\ 3,\ 4

    Because the common difference is 1, the nth term must start with n. Working out n for n=1n = 1, 2, 3, 4 gives 1, 2, 3, 4.

  3. Adjust by the constant and write the nth term

    n gives 1, 2, 3, 4; add 3: n+3n \text{ gives } 1,\ 2,\ 3,\ 4; \text{ add } 3 \text{: } n + 3

    Each term of the sequence is 3 more than the matching multiple of 1, so the nth term is n + 3. Check n=1n = 1: 1+3=1+3=41 + 3 = 1 + 3 = 4, the first term.

Answer
n+3n + 3

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