Recurrence and general sequences Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Recurrence and general sequences questions. See exactly how to solve problems on recurrence, sequences, series, limit.

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A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A sequence is defined by un+1=2un+3u_{n+1}=2 u_{n} + 3 with u1=1u_1=1. Find u2u_{2}.

Worked solution

  1. Write down the recurrence and the first term

    un+1=2un+3,u1=1u_{n+1}=2 u_{n} + 3,\quad u_1=1

    Each term is generated from the one before it.

  2. Compute the term u_2

    u2=2×1+3=5u_{2}=2\times 1+3=5

    Apply the recurrence relation to the previous term.

  3. State the required term

    u2=5u_{2}=5

    This is the value that was asked for.

Answer
55
Question 2
2 markseasy
A sequence is defined by un+1=3un−1u_{n+1}=3 u_{n} - 1 with u1=2u_1=2. Find u2u_{2}.

Worked solution

  1. Write down the recurrence and the first term

    un+1=3un−1,u1=2u_{n+1}=3 u_{n} - 1,\quad u_1=2

    Each term is generated from the one before it.

  2. Compute the term u_2

    u2=3×2−1=5u_{2}=3\times 2-1=5

    Apply the recurrence relation to the previous term.

  3. State the required term

    u2=5u_{2}=5

    This is the value that was asked for.

Answer
55
Question 3
2 markseasy
A sequence is defined by un+1=un+4u_{n+1}=u_{n} + 4 with u1=7u_1=7. Find u2u_{2}.

Worked solution

  1. Write down the recurrence and the first term

    un+1=un+4,u1=7u_{n+1}=u_{n} + 4,\quad u_1=7

    Each term is generated from the one before it.

  2. Compute the term u_2

    u2=7+4=11u_{2}=7+4=11

    Apply the recurrence relation to the previous term.

  3. State the required term

    u2=11u_{2}=11

    This is the value that was asked for.

Answer
1111
Question 4
2 markseasy
A sequence is defined by un+1=5−2unu_{n+1}=5 - 2 u_{n} with u1=3u_1=3. Find u2u_{2}.

Worked solution

  1. Write down the recurrence and the first term

    un+1=5−2un,u1=3u_{n+1}=5 - 2 u_{n},\quad u_1=3

    Each term is generated from the one before it.

  2. Compute the term u_2

    u2=−2×3+5=−1u_{2}=-2\times 3+5=-1

    Apply the recurrence relation to the previous term.

  3. State the required term

    u2=−1u_{2}=-1

    This is the value that was asked for.

Answer
−1-1
Question 5
2 markseasy
A sequence is defined by un+1=4unu_{n+1}=4 u_{n} with u1=2u_1=2. Find u2u_{2}.

Worked solution

  1. Write down the recurrence and the first term

    un+1=4un,u1=2u_{n+1}=4 u_{n},\quad u_1=2

    Each term is generated from the one before it.

  2. Compute the term u_2

    u2=4×2=8u_{2}=4\times 2=8

    Apply the recurrence relation to the previous term.

  3. State the required term

    u2=8u_{2}=8

    This is the value that was asked for.

Answer
88

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