Recurrence relations Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Recurrence relations questions. See exactly how to solve problems on recurrence-relations, first-order, modelling, tower-of-hanoi.

recurrence-relationsfirst-ordermodellingtower-of-hanoipopulationdepreciation
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
In the Tower of Hanoi puzzle, unu_{n} is the least number of single-disc moves needed to transfer a tower of nn discs. The model is un+1=2un+1u_{n+1}=2u_{n}+1 for n≥1n\ge 1, with u1=1u_{1}=1. Find the least number of moves needed for a tower of 66 discs, that is the value of u6u_{6}.

Worked solution

  1. Apply the model with n=1n=1

    u2=2⋅1+(1)=3u_{2}=2\cdot 1+\left(1\right)=3

    Each new value comes straight from the model applied to the value just before it.

  2. Apply the model with n=2n=2

    u3=2⋅3+(1)=7u_{3}=2\cdot 3+\left(1\right)=7

    Each new value comes straight from the model applied to the value just before it.

  3. State the required value

    u6=63u_{6}=63

    This is the quantity the model is asked for.

Answer
6363
Question 2
2 markseasy
A colony of bacteria triples each hour and then 44 hundred are removed for testing; unu_{n} is the colony size, in hundreds, after nn hours. The model is un+1=3un−4u_{n+1}=3u_{n}-4 for n≥0n\ge 0, with u0=5u_{0}=5. Find the colony size after 44 hours, that is the value of u4u_{4}.

Worked solution

  1. Apply the model with n=0n=0

    u1=3⋅5+(−4)=11u_{1}=3\cdot 5+\left(-4\right)=11

    Each new value comes straight from the model applied to the value just before it.

  2. Apply the model with n=1n=1

    u2=3⋅11+(−4)=29u_{2}=3\cdot 11+\left(-4\right)=29

    Each new value comes straight from the model applied to the value just before it.

  3. List the values found so far

    u0=5, u1=11, u2=29, u3=83, u4=245u_{0}=5,\ u_{1}=11,\ u_{2}=29,\ u_{3}=83,\ u_{4}=245

    Writing the run of values in order makes an arithmetic slip easy to spot.

  4. State the required value

    u4=245u_{4}=245

    This is the quantity the model is asked for.

Answer
245245
Question 3
2 markseasy
A drone's altitude, in metres, is halved every second as it descends; unu_{n} is the altitude after nn seconds. The model is un+1=12unu_{n+1}=\frac{1}{2}u_{n} for n≥0n\ge 0, with u0=320u_{0}=320. Find the altitude after 55 seconds, that is the value of u5u_{5}.

Worked solution

  1. Apply the model with n=0n=0

    u1=12⋅320=160u_{1}=\frac{1}{2}\cdot 320=160

    Each new value comes straight from the model applied to the value just before it.

  2. Apply the model with n=1n=1

    u2=12⋅160=80u_{2}=\frac{1}{2}\cdot 160=80

    Each new value comes straight from the model applied to the value just before it.

  3. State the required value

    u5=10u_{5}=10

    This is the quantity the model is asked for.

Answer
1010
Question 4
2 markseasy
A savings tin gains £7\pounds7 each week with no interest; unu_{n} is the amount, in pounds, after nn weeks. The model is un+1=un+7u_{n+1}=u_{n}+7 for n≥0n\ge 0, with u0=12u_{0}=12. Find the amount saved after 88 weeks, that is the value of u8u_{8}.

Worked solution

  1. Apply the model with n=0n=0

    u1=12+(7)=19u_{1}=12+\left(7\right)=19

    Each new value comes straight from the model applied to the value just before it.

  2. Apply the model with n=1n=1

    u2=19+(7)=26u_{2}=19+\left(7\right)=26

    Each new value comes straight from the model applied to the value just before it.

  3. List the values found so far

    u0=12, u1=19, u2=26, u3=33, u4=40u_{0}=12,\ u_{1}=19,\ u_{2}=26,\ u_{3}=33,\ u_{4}=40

    Writing the run of values in order makes an arithmetic slip easy to spot.

  4. State the required value

    u8=68u_{8}=68

    This is the quantity the model is asked for.

Answer
6868
Question 5
2 markseasy
In a stepping-stone game a token doubles its score and then loses 33 points each turn; unu_{n} is the score after nn turns. The model is un+1=2un−3u_{n+1}=2u_{n}-3 for n≥0n\ge 0, with u0=10u_{0}=10. Find the score after 55 turns, that is the value of u5u_{5}.

Worked solution

  1. Apply the model with n=0n=0

    u1=2⋅10+(−3)=17u_{1}=2\cdot 10+\left(-3\right)=17

    Each new value comes straight from the model applied to the value just before it.

  2. Apply the model with n=1n=1

    u2=2⋅17+(−3)=31u_{2}=2\cdot 17+\left(-3\right)=31

    Each new value comes straight from the model applied to the value just before it.

  3. State the required value

    u5=227u_{5}=227

    This is the quantity the model is asked for.

Answer
227227

Unlock 65 more Recurrence relations questions

Create a free account to work through every Further Maths Recurrence relations question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Recurrence relations practice

Hard QuestionsPractice Questions

Related Decision Maths topics

AlgorithmsAllocation (assignment)Critical path analysisDecision analysisDynamic programmingFlows in networks