GCSE Generating sequences Practice Questions

Free GCSE Generating sequences practice questions with full step-by-step worked solutions. Covers term-to-term rule, generating terms, continuing a sequence, position-to-term rule. Practise exam-style problems and check your method.

term-to-term rulegenerating termscontinuing a sequenceposition-to-term rulenth term substitutiondecreasing sequence
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
The first term of a sequence is 55. The term-to-term rule is 'add 44'. Write down the second term.
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Worked solution

  1. Write down the first term

    a1=5a_1 = 5

    The sequence starts at 5.

  2. Apply the term-to-term rule 'add 4'

    a2=5+4a_2 = 5 + 4

    A term-to-term rule tells you how to get from one term to the NEXT one: add 4 to the term before.

  3. Work out the second term

    a2=9a_2 = 9

    5 add 4 is 9, so the second term is 9.

Answer
99
Question 2
2 markseasy
The first term of a sequence is 1.51.5. The term-to-term rule is 'add 0.50.5'. Work out the third term.
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Worked solution

  1. Write down the first term

    a1=1.5a_1 = 1.5

    The sequence starts at 1.5.

  2. Find the second term

    a2=1.5+0.5=2a_2 = 1.5 + 0.5 = 2

    Adding 0.5 to 1.5 gives 2.

  3. Find the third term

    a3=2+0.5=2.5a_3 = 2 + 0.5 = 2.5

    Add 0.5 once more to get the third term.

Answer
2.52.5
Question 3
2 marksintermediate
The nnth term of a sequence is 3n−13n - 1. Is 2525 a term of this sequence? Give a reason for your answer.
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Worked solution

  1. Set the rule equal to 25

    3n−1=253n - 1 = 25

    If 25 is a term, some whole-number position n must give it.

  2. Add 1 to both sides

    3n=263n = 26

    Undo the -1.

  3. Divide both sides by 3

    n=263=8.66…n = \frac{26}{3} = 8.66\ldots

    26 does not divide exactly by 3.

  4. Interpret n

    n must be a positive whole numbern \text{ must be a positive whole number}

    There is no such thing as the 8.67th term, so 25 cannot appear.

  5. Check the terms either side

    3×8−1=23,3×9−1=263 \times 8 - 1 = 23, \quad 3 \times 9 - 1 = 26

    The 8th term is 23 and the 9th is 26; the sequence jumps straight over 25.

  6. Conclude

    25 is not a term25 \text{ is not a term}

    Because n came out as a fraction, 25 is not in the sequence.

Answer
No: 3n−1=25 gives n=263, not a whole number\text{No: } 3n - 1 = 25 \text{ gives } n = \tfrac{26}{3}, \text{ not a whole number}
Question 4
4 markshard
Sequence P has first term 22 and term-to-term rule 'add 66'. The nnth term of sequence Q is 4n−24n - 2. The 1010th term of Q is also a term of P. Work out its position in sequence P.
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Worked solution

  1. Write down Q's rule

    4n−24n - 2

    Q is given by a position-to-term rule.

  2. Find Q's 10th term

    4×10−2=384 \times 10 - 2 = 38

    Substitute n=10n = 10: 40−2=3840 - 2 = 38.

  3. Write down P's rule

    p1=2,add 6p_1 = 2, \quad \text{add } 6

    P is given by a term-to-term rule.

  4. Generate the first four terms of P

    2, 8, 14, 202,\ 8,\ 14,\ 20

    2, 8, 14, 20.

  5. Generate three more terms of P

    26, 32, 3826,\ 32,\ 38

    26, 32, 38.

  6. Spot 38 in sequence P

    3838

    38 does appear in P.

  7. Work out the total increase

    38−2=3638 - 2 = 36

    From the first term of P up to 38 is a rise of 36.

  8. Count the jumps

    36÷6=6 jumps36 \div 6 = 6 \text{ jumps}

    Each jump adds 6.

  9. Turn jumps into a position

    1+6=71 + 6 = 7

    Six jumps AFTER the 1st term lands on the 7th term — this is where off-by-one errors happen.

  10. State the position

    n=7n = 7

    38 is the 7th term of sequence P.

Answer
n=7n = 7
Question 5
6 markschallenging
The nnth term of a sequence is 2n+52n + 5. Sam says, "Every term of this sequence is odd." Is Sam right? Explain your answer, and work out the 5050th term.
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Worked solution

  1. Generate the first term

    2×1+5=72 \times 1 + 5 = 7

    The first term is 7, which is odd.

  2. Generate the second term

    2×2+5=92 \times 2 + 5 = 9

    The second term is 9, also odd.

  3. Generate the third term

    2×3+5=112 \times 3 + 5 = 11

    The third term is 11, odd again.

  4. Notice that examples are not a proof

    7, 9, 11, …7,\ 9,\ 11,\ \ldots

    Three odd terms do not prove that ALL of them are odd — you need a reason.

  5. Look at the first part of the rule

    2n is always even2n \text{ is always even}

    2 multiplied by any whole number gives an even number.

  6. Add the 5

    even+5=odd\text{even} + 5 = \text{odd}

    An even number plus an odd number is always odd.

  7. Answer the first part

    Sam is right\text{Sam is right}

    Every term of the sequence is odd.

  8. Start the 50th term

    2×50=1002 \times 50 = 100

    Substitute n=50n = 50 into the rule.

  9. Finish the 50th term

    100+5=105100 + 5 = 105

    The 50th term is 105.

  10. Check it against the rule

    105 is odd105 \text{ is odd}

    Consistent with the argument above.

  11. Name the common error

    n=50 is even, so the term must be evenn = 50 \text{ is even, so the term must be even}

    This is the trap: students often assume the term copies the parity of the position.

  12. Explain why that is wrong

    the +5 changes the parity\text{the } +5 \text{ changes the parity}

    2n is even whatever n is, so the +5 always tips the term to odd.

  13. Test an odd position too

    2×51+5=1072 \times 51 + 5 = 107

    The 51st term is odd as well — the parity of n makes no difference.

  14. Describe the sequence

    7, 9, 11, 13, …7,\ 9,\ 11,\ 13,\ \ldots

    The sequence is simply the odd numbers from 7 upwards.

  15. State the full answer

    Sam is right; the 50th term is 105\text{Sam is right; the 50th term is } 105

    Every term is odd, and the 50th term is 105.

Answer
Sam is right; the 50th term is 105\text{Sam is right; the 50th term is } 105

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