Hard A-Level Recurrence and general sequences Questions

Challenging, exam-style A-Level Recurrence and general sequences questions with worked solutions. Stretch yourself on the hardest recurrence, sequences, series, limit problems.

recurrencesequencesserieslimitperiodicconstants
A-Level34 questionsStep-by-step solutions
Question 1
8 markschallenging
The sequence defined by un+1=−1unu_{n+1}=-\dfrac{1}{u_{n}} with u1=4u_1=4 is best described as:
Show worked solution

Worked solution

  1. Write down the recurrence and the first term

    un+1=−1un,u1=4u_{n+1}=-\dfrac{1}{u_{n}},\quad u_1=4

    The recurrence defines the whole sequence.

  2. Compute the term u_2

    u2=−14=−14u_{2}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=−1(−14)=4u_{3}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=−14=−14u_{4}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  5. Compute the term u_5

    u5=−1(−14)=4u_{5}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  6. Compute the term u_6

    u6=−14=−14u_{6}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  7. Compute the term u_7

    u7=−1(−14)=4u_{7}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  8. Compute the term u_8

    u8=−14=−14u_{8}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  9. Compute the term u_9

    u9=−1(−14)=4u_{9}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  10. Compute the term u_10

    u10=−14=−14u_{10}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  11. Compute the term u_11

    u11=−1(−14)=4u_{11}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  12. Compute the term u_12

    u12=−14=−14u_{12}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  13. Compute the term u_13

    u13=−1(−14)=4u_{13}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  14. Examine the behaviour of the terms

    u1=4, u2=−14, u3=4, u4=−14, u5=4u_{1}=4,\ u_{2}=- \frac{1}{4},\ u_{3}=4,\ u_{4}=- \frac{1}{4},\ u_{5}=4

    Looking at successive terms reveals the pattern.

  15. State the classification

    a periodic sequence\text{a periodic sequence}

    This describes the long-term behaviour of the sequence.

Answer
a periodic sequence\text{a periodic sequence}
Question 2
8 markschallenging
The sequence defined by un+1=11−unu_{n+1}=\dfrac{1}{1-u_{n}} with u1=2u_1=2 is best described as:
Show worked solution

Worked solution

  1. Write down the recurrence and the first term

    un+1=11−un,u1=2u_{n+1}=\dfrac{1}{1-u_{n}},\quad u_1=2

    The recurrence defines the whole sequence.

  2. Compute the term u_2

    u2=11−2=−1u_{2}=\dfrac{1}{1-2}=-1

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=11−(−1)=12u_{3}=\dfrac{1}{1-\left(-1\right)}=\frac{1}{2}

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=11−12=2u_{4}=\dfrac{1}{1-\frac{1}{2}}=2

    Apply the recurrence relation to the previous term.

  5. Compute the term u_5

    u5=11−2=−1u_{5}=\dfrac{1}{1-2}=-1

    Apply the recurrence relation to the previous term.

  6. Compute the term u_6

    u6=11−(−1)=12u_{6}=\dfrac{1}{1-\left(-1\right)}=\frac{1}{2}

    Apply the recurrence relation to the previous term.

  7. Compute the term u_7

    u7=11−12=2u_{7}=\dfrac{1}{1-\frac{1}{2}}=2

    Apply the recurrence relation to the previous term.

  8. Compute the term u_8

    u8=11−2=−1u_{8}=\dfrac{1}{1-2}=-1

    Apply the recurrence relation to the previous term.

  9. Compute the term u_9

    u9=11−(−1)=12u_{9}=\dfrac{1}{1-\left(-1\right)}=\frac{1}{2}

    Apply the recurrence relation to the previous term.

  10. Compute the term u_10

    u10=11−12=2u_{10}=\dfrac{1}{1-\frac{1}{2}}=2

    Apply the recurrence relation to the previous term.

  11. Compute the term u_11

    u11=11−2=−1u_{11}=\dfrac{1}{1-2}=-1

    Apply the recurrence relation to the previous term.

  12. Compute the term u_12

    u12=11−(−1)=12u_{12}=\dfrac{1}{1-\left(-1\right)}=\frac{1}{2}

    Apply the recurrence relation to the previous term.

  13. Compute the term u_13

    u13=11−12=2u_{13}=\dfrac{1}{1-\frac{1}{2}}=2

    Apply the recurrence relation to the previous term.

  14. Examine the behaviour of the terms

    u1=2, u2=−1, u3=12, u4=2, u5=−1u_{1}=2,\ u_{2}=-1,\ u_{3}=\frac{1}{2},\ u_{4}=2,\ u_{5}=-1

    Looking at successive terms reveals the pattern.

  15. State the classification

    a periodic sequence\text{a periodic sequence}

    This describes the long-term behaviour of the sequence.

Answer
a periodic sequence\text{a periodic sequence}
Question 3
8 markschallenging
The sequence defined by un+1=un−6u_{n+1}=u_{n} - 6 with u1=100u_1=100 is best described as:
Show worked solution

Worked solution

  1. Write down the recurrence and the first term

    un+1=un−6,u1=100u_{n+1}=u_{n} - 6,\quad u_1=100

    The recurrence defines the whole sequence.

  2. Compute the term u_2

    u2=100−6=94u_{2}=100-6=94

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=94−6=88u_{3}=94-6=88

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=88−6=82u_{4}=88-6=82

    Apply the recurrence relation to the previous term.

  5. Compute the term u_5

    u5=82−6=76u_{5}=82-6=76

    Apply the recurrence relation to the previous term.

  6. Compute the term u_6

    u6=76−6=70u_{6}=76-6=70

    Apply the recurrence relation to the previous term.

  7. Compute the term u_7

    u7=70−6=64u_{7}=70-6=64

    Apply the recurrence relation to the previous term.

  8. Compute the term u_8

    u8=64−6=58u_{8}=64-6=58

    Apply the recurrence relation to the previous term.

  9. Compute the term u_9

    u9=58−6=52u_{9}=58-6=52

    Apply the recurrence relation to the previous term.

  10. Compute the term u_10

    u10=52−6=46u_{10}=52-6=46

    Apply the recurrence relation to the previous term.

  11. Compute the term u_11

    u11=46−6=40u_{11}=46-6=40

    Apply the recurrence relation to the previous term.

  12. Compute the term u_12

    u12=40−6=34u_{12}=40-6=34

    Apply the recurrence relation to the previous term.

  13. Compute the term u_13

    u13=34−6=28u_{13}=34-6=28

    Apply the recurrence relation to the previous term.

  14. Examine the behaviour of the terms

    u1=100, u2=94, u3=88, u4=82, u5=76u_{1}=100,\ u_{2}=94,\ u_{3}=88,\ u_{4}=82,\ u_{5}=76

    Looking at successive terms reveals the pattern.

  15. State the classification

    a decreasing sequence\text{a decreasing sequence}

    This describes the long-term behaviour of the sequence.

Answer
a decreasing sequence\text{a decreasing sequence}
Question 4
8 markschallenging
The sequence defined by un+1=un+9u_{n+1}=u_{n} + 9 with u1=0u_1=0 is best described as:
Show worked solution

Worked solution

  1. Write down the recurrence and the first term

    un+1=un+9,u1=0u_{n+1}=u_{n} + 9,\quad u_1=0

    The recurrence defines the whole sequence.

  2. Compute the term u_2

    u2=0+9=9u_{2}=0+9=9

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=9+9=18u_{3}=9+9=18

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=18+9=27u_{4}=18+9=27

    Apply the recurrence relation to the previous term.

  5. Compute the term u_5

    u5=27+9=36u_{5}=27+9=36

    Apply the recurrence relation to the previous term.

  6. Compute the term u_6

    u6=36+9=45u_{6}=36+9=45

    Apply the recurrence relation to the previous term.

  7. Compute the term u_7

    u7=45+9=54u_{7}=45+9=54

    Apply the recurrence relation to the previous term.

  8. Compute the term u_8

    u8=54+9=63u_{8}=54+9=63

    Apply the recurrence relation to the previous term.

  9. Compute the term u_9

    u9=63+9=72u_{9}=63+9=72

    Apply the recurrence relation to the previous term.

  10. Compute the term u_10

    u10=72+9=81u_{10}=72+9=81

    Apply the recurrence relation to the previous term.

  11. Compute the term u_11

    u11=81+9=90u_{11}=81+9=90

    Apply the recurrence relation to the previous term.

  12. Compute the term u_12

    u12=90+9=99u_{12}=90+9=99

    Apply the recurrence relation to the previous term.

  13. Compute the term u_13

    u13=99+9=108u_{13}=99+9=108

    Apply the recurrence relation to the previous term.

  14. Examine the behaviour of the terms

    u1=0, u2=9, u3=18, u4=27, u5=36u_{1}=0,\ u_{2}=9,\ u_{3}=18,\ u_{4}=27,\ u_{5}=36

    Looking at successive terms reveals the pattern.

  15. State the classification

    an increasing sequence\text{an increasing sequence}

    This describes the long-term behaviour of the sequence.

Answer
an increasing sequence\text{an increasing sequence}
Question 5
8 markschallenging
A sequence satisfies un+1=a un+bu_{n+1}=a\,u_n+b. Given that u1=1u_1=1, u2=2u_2=2 and u3=1u_3=1, find the values of aa and bb.
Show worked solution

Worked solution

  1. Note the three given terms

    u1=1, u2=2, u3=1u_1=1,\ u_2=2,\ u_3=1

    Three terms provide two simultaneous equations.

  2. Form the first equation

    u2=a u1+b:2=1 a+bu_2=a\,u_1+b:\quad 2=1\,a+b

    Substitute n=1 into the recurrence.

  3. Form the second equation

    u3=a u2+b:1=2 a+bu_3=a\,u_2+b:\quad 1=2\,a+b

    Substitute n=2 into the recurrence.

  4. Subtract the equations to eliminate b

    −1=1 a ⇒ a=−1-1=1\,a\ \Rightarrow\ a=-1

    Subtracting removes b and leaves a.

  5. Back-substitute to find b

    b=2−−1×1=3b=2--1\times 1=3

    Use either equation with the known a.

  6. Check the term u_4

    u4=−1+3=2u_{4}=-1+3=2

    Apply the recurrence relation to the previous term.

  7. Check the term u_5

    u5=−2+3=1u_{5}=-2+3=1

    Apply the recurrence relation to the previous term.

  8. Check the term u_6

    u6=−1+3=2u_{6}=-1+3=2

    Apply the recurrence relation to the previous term.

  9. Check the term u_7

    u7=−2+3=1u_{7}=-2+3=1

    Apply the recurrence relation to the previous term.

  10. Check the term u_8

    u8=−1+3=2u_{8}=-1+3=2

    Apply the recurrence relation to the previous term.

  11. Check the term u_9

    u9=−2+3=1u_{9}=-2+3=1

    Apply the recurrence relation to the previous term.

  12. Check the term u_10

    u10=−1+3=2u_{10}=-1+3=2

    Apply the recurrence relation to the previous term.

  13. Check the term u_11

    u11=−2+3=1u_{11}=-2+3=1

    Apply the recurrence relation to the previous term.

  14. Check the term u_12

    u12=−1+3=2u_{12}=-1+3=2

    Apply the recurrence relation to the previous term.

  15. State the values of a and b

    a=−1, b=3a=-1,\ b=3

    These constants reproduce all the given terms.

Answer
a=−1, b=3a=-1,\ b=3

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