Two related arithmetic progressions

Two arithmetic progressions are related by given conditions; form simultaneous equations to find their parameters.

13 questions · Moderate -0.1

1.04h Arithmetic sequences: nth term and sum formulae
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CAIE P1 2021 June Q9
9 marks Standard +0.3
9
  1. A geometric progression is such that the second term is equal to \(24 \%\) of the sum to infinity. Find the possible values of the common ratio.
  2. An arithmetic progression \(P\) has first term \(a\) and common difference \(d\). An arithmetic progression \(Q\) has first term 2( \(a + 1\) ) and common difference ( \(d + 1\) ). It is given that $$\frac { 5 \text { th term of } P } { 12 \text { th term of } Q } = \frac { 1 } { 3 } \quad \text { and } \quad \frac { \text { Sum of first } 5 \text { terms of } P } { \text { Sum of first } 5 \text { terms of } Q } = \frac { 2 } { 3 } .$$ Find the value of \(a\) and the value of \(d\).
Edexcel C1 2010 June Q9
8 marks Moderate -0.8
  1. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays \(\pounds a\) for their first day, \(\pounds ( a + d )\) for their second day, \(\pounds ( a + 2 d )\) for their third day, and so on, thus increasing the daily payment by \(\pounds d\) for each extra day they work.
A picker who works for all 30 days will earn \(\pounds 40.75\) on the final day.
  1. Use this information to form an equation in \(a\) and \(d\). A picker who works for all 30 days will earn a total of \(\pounds 1005\)
  2. Show that \(15 ( a + 40.75 ) = 1005\)
  3. Hence find the value of \(a\) and the value of \(d\).
Edexcel C1 2014 June Q10
8 marks Moderate -0.8
  1. Xin has been given a 14 day training schedule by her coach.
Xin will run for \(A\) minutes on day 1 , where \(A\) is a constant.
She will then increase her running time by ( \(d + 1\) ) minutes each day, where \(d\) is a constant.
  1. Show that on day 14 , Xin will run for $$( A + 13 d + 13 ) \text { minutes. }$$ Yi has also been given a 14 day training schedule by her coach.
    Yi will run for \(( A - 13 )\) minutes on day 1 .
    She will then increase her running time by ( \(2 d - 1\) ) minutes each day.
    Given that Yi and Xin will run for the same length of time on day 14,
  2. find the value of \(d\). Given that Xin runs for a total time of 784 minutes over the 14 days,
  3. find the value of \(A\).
Edexcel AEA 2010 June Q2
11 marks Challenging +1.2
2.The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ,where \(p\) and \(q\) are positive integers and \(p \neq q\) . Giving simplified answers in terms of \(p\) and \(q\) ,find
  1. the common difference of the terms in this series,
  2. the first term of the series,
  3. the sum of the first \(( p + q )\) terms of the series.
Edexcel C1 Q9
11 marks Standard +0.3
  1. The second and fifth terms of an arithmetic series are 26 and 41 repectively.
    1. Show that the common difference of the series is 5 .
    2. Find the 12th term of the series.
    Another arithmetic series has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two series are equal,
  2. find the value of \(n\).
OCR C2 Q7
10 marks Standard +0.3
  1. Show that the common difference is 5 .
  2. Find the 12th term. Another arithmetic sequence has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two sequences are equal,
  3. find the value of \(n\).
AQA Paper 1 2021 June Q6
7 marks Standard +0.3
6
  1. The ninth term of an arithmetic series is 3 The sum of the first \(n\) terms of the series is \(S _ { n }\) and \(S _ { 21 } = 42\) Find the first term and common difference of the series.
    [0pt] [4 marks]
    6
  2. A second arithmetic series has first term - 18 and common difference \(\frac { 3 } { 4 }\) The sum of the first \(n\) terms of this series is \(T _ { n }\) Find the value of \(n\) such that \(T _ { n } = S _ { n }\) [0pt] [3 marks]
Edexcel C1 Q7
8 marks Moderate -0.8
Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays £500. Her payments then increase by £50 each year, so that she pays £550 in the second year, £600 in the third year, and so on.
  1. Find the amount that Anne will pay in the 40th year. [2]
  2. Find the total amount that Anne will pay in over the 40 years. [2]
Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
  1. find the value of \(d\). [4]
Edexcel C1 Specimen Q7
9 marks Moderate -0.8
Ahmed plans to save £250 in the year 2001, £300 in 2002, £350 in 2003, and so on until the year 2020. His planned savings form an arithmetic sequence with common difference £50.
  1. Find the amount he plans to save in the year 2011. [2]
  2. Calculate his total planned savings over the 20 year period from 2001 to 2020. [3]
Ben also plans to save money over the same 20 year period. He saves £\(A\) in the year 2001 and his planned yearly savings form an arithmetic sequence with common difference £60. Given that Ben's total planned savings over the 20 year period are equal to Ahmed's total planned savings over the same period,
  1. calculate the value of \(A\). [4]
Edexcel C1 Q6
8 marks Moderate -0.8
Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays £500. Her payments then increase by £50 each year, so that she pays £550 in the second year, £600 in the third year, and so on.
  1. Find the amount that Anne will pay in the 40th year. [2]
  2. Find the total amount that Anne will pay in over the 40 years. [2]
Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
  1. find the value of \(d\). [4]
OCR MEI C2 2013 January Q11
12 marks Moderate -0.3
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250.
    1. Find the values of \(A\) and \(D\). [4]
    2. Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250. Use the formula for the sum of a geometric progression to show that \(\frac{r^4 - 1}{r^2 - 1} = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\). [5]
OCR MEI C2 Q1
12 marks Standard +0.3
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250.
    1. Find the values of \(A\) and \(D\). [4]
    2. Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250. Use the formula for the sum of a geometric progression to show that \(\frac{r^4 - 1}{r^2 - 1} = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\). [5]
Pre-U Pre-U 9794/2 2010 June Q3
6 marks Standard +0.3
An arithmetic progression has 13th term equal to 60 and 31st term equal to 141.
  1. Find the first term and common difference of the progression. [3]
A second arithmetic progression has first term 1.5 and common difference 3.
    1. Write down the first four terms of each progression. [1]
    2. Prove that the two progressions have an infinite number of terms in common. [2]