Geometric sequences and series Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Geometric sequences and series questions. See exactly how to solve problems on geometric-sequence, nth-term, common-ratio, geometric-series.

geometric-sequencenth-termcommon-ratiogeometric-seriessum-of-n-termssum-to-infinity
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A geometric sequence has first term 22 and common ratio 33. Find the 55th term u5u_{5}.

Worked solution

  1. Identify the first term

    a=2a=2

    The first term of the sequence.

  2. Identify the common ratio

    r=3r=3

    The constant multiplier between terms.

  3. State the required term

    u5=162u_{5}=162

    This is the value of the requested term.

Answer
162162
Question 2
2 markseasy
A geometric sequence has first term 55 and common ratio 22. Find the 44th term u4u_{4}.

Worked solution

  1. Identify the first term

    a=5a=5

    The first term of the sequence.

  2. Identify the common ratio

    r=2r=2

    The constant multiplier between terms.

  3. State the required term

    u4=40u_{4}=40

    This is the value of the requested term.

Answer
4040
Question 3
2 markseasy
A geometric sequence has first term 33 and common ratio 22. Find the 66th term u6u_{6}.

Worked solution

  1. Identify the first term

    a=3a=3

    The first term of the sequence.

  2. Identify the common ratio

    r=2r=2

    The constant multiplier between terms.

  3. State the required term

    u6=96u_{6}=96

    This is the value of the requested term.

Answer
9696
Question 4
2 markseasy
The first three terms of a geometric sequence are 22, 66 and 1818. Find the common ratio rr.

Worked solution

  1. Write the given terms

    2, 6, 182,\ 6,\ 18

    The first three terms of the sequence.

  2. Recall how to find the common ratio

    r=un+1unr=\frac{u_{n+1}}{u_n}

    The common ratio is the quotient of successive terms.

  3. Write down the common ratio

    r=3\therefore r=3

    This is the common ratio of the sequence.

Answer
33
Question 5
2 markseasy
The first three terms of a geometric sequence are 55, 1010 and 2020. Find the common ratio rr.

Worked solution

  1. Write the given terms

    5, 10, 205,\ 10,\ 20

    The first three terms of the sequence.

  2. Recall how to find the common ratio

    r=un+1unr=\frac{u_{n+1}}{u_n}

    The common ratio is the quotient of successive terms.

  3. Write down the common ratio

    r=2\therefore r=2

    This is the common ratio of the sequence.

Answer
22

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