Sigma notation: arithmetic series evaluation

Evaluate or simplify a sum in sigma notation by recognising it as an arithmetic series and applying the standard sum formula, including finding first term, common difference, or sum for given n.

11 questions · Easy -1.1

1.04h Arithmetic sequences: nth term and sum formulae
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Edexcel C12 2014 June Q9
7 marks Moderate -0.8
9. (i) Find the value of \(\sum _ { r = 1 } ^ { 20 } ( 3 + 5 r )\) (ii) Given that \(\sum _ { r = 0 } ^ { \infty } \frac { a } { 4 ^ { r } } = 16\), find the value of the constant \(a\).
OCR C2 Q3
7 marks Moderate -0.8
  1. (i) Evaluate
$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.
OCR MEI C2 Q3
5 marks Easy -1.2
3 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.
OCR MEI C2 2012 June Q2
4 marks Easy -1.8
2 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5 \\ & u _ { n + 1 } = u _ { n } + 3 . \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.
OCR MEI Paper 2 2024 June Q7
6 marks Easy -1.3
7 A sequence is defined by the recurrence relation \(\mathrm { u } _ { \mathrm { k } + 1 } = \mathrm { u } _ { \mathrm { k } } + 5\) with \(\mathrm { u } _ { 1 } = - 2\).
  1. Write down the values of \(u _ { 2 } , u _ { 3 }\), and \(u _ { 4 }\).
  2. Explain whether this sequence is divergent or convergent.
  3. Determine the value of \(u _ { 30 }\).
  4. Determine the value of \(\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }\).
Edexcel C1 Q2
6 marks Easy -1.2
2. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r ) .$$
  1. Write down the first two terms of the series.
  2. Find the common difference of the series. Given that \(n = 50\),
  3. find the sum of the series.
AQA Paper 1 2020 June Q10
12 marks Moderate -0.8
10
  1. An arithmetic series is given by $$\sum _ { r = 5 } ^ { 20 } ( 4 r + 1 )$$ 10
    1. (i) Write down the first term of the series.
      10
    2. (ii) Write down the common difference of the series.
      10
    3. (iii) Find the number of terms of the series.
      10
    4. A different arithmetic series is given by \(\sum _ { r = 10 } ^ { 100 } ( b r + c )\)
      where \(b\) and \(c\) are constants.
      The sum of this series is 7735
      10
    5. (ii) The 40th term of the series is 4 times the 2nd term. Find the values of \(b\) and \(c\).
      [0pt] [4 marks]
Edexcel C1 Q29
6 marks Easy -1.2
The sum of an arithmetic series is $$\sum_{r=1}^{n} (80 - 3r).$$
  1. Write down the first two terms of the series. [2]
  2. Find the common difference of the series. [1]
Given that \(n = 50\),
  1. find the sum of the series. [3]
Edexcel C1 Specimen Q1
3 marks Easy -1.2
Calculate \(\sum_{r=1}^{20} 5 + 2r\) [3]
Edexcel C1 Q1
3 marks Easy -1.2
Evaluate $$\sum_{r=1}^{20} (3r + 4).$$ [3]
Edexcel C1 Q6
7 marks Moderate -0.8
  1. Evaluate $$\sum_{r=1}^{50} (80 - 3r).$$ [3]
  2. Show that $$\sum_{r=1}^{n} \frac{r + 3}{2} = k n(n + 7),$$ where \(k\) is a rational constant to be found. [4]