Sigma notation: direct numerical evaluation

Evaluate a sum given in sigma notation by computing each term and adding, where the expression is not a standard arithmetic or geometric series (e.g., rational, cubic, or mixed terms).

18 questions · Easy -1.2

1.04g Sigma notation: for sums of series
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Edexcel C1 2014 January Q5
5 marks Moderate -0.8
5. Given that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } a _ { r } = 12 + 4 n ^ { 2 }$$
  1. find the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\)
  2. Find the value of \(a _ { 6 }\)
OCR MEI C2 2006 January Q2
2 marks Easy -1.8
2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).
OCR MEI C2 2007 June Q4
4 marks Easy -1.3
4
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 2009 June Q3
3 marks Moderate -0.8
3
  1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
  2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.
OCR MEI C2 Q7
4 marks Easy -1.3
7
  1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.
OCR MEI C2 Q8
3 marks Moderate -0.8
8
  1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
  2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.
OCR MEI C2 Q9
4 marks Easy -1.2
9
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 Q11
2 marks Easy -1.2
11 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).
OCR MEI C2 Q1
2 marks Easy -1.2
1 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 Q5
2 marks Easy -1.8
5 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).
OCR MEI C2 2009 January Q3
2 marks Easy -1.8
3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 2011 January Q1
2 marks Easy -1.2
1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).
OCR MEI C2 2012 January Q1
2 marks Easy -1.2
1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).
OCR MEI Paper 3 2020 November Q1
2 marks Easy -1.2
1 Find the value of \(\sum _ { r = 1 } ^ { 5 } 2 ^ { r } ( r - 1 )\).
OCR MEI Paper 3 2021 November Q6
4 marks Moderate -0.8
6 In this question you must show detailed reasoning.
Show that \(\sum _ { r = 1 } ^ { 3 } \frac { 1 } { \sqrt { r + 1 } + \sqrt { r } } = 1\).
OCR MEI C2 2010 June Q2
4 marks Moderate -0.8
  1. Evaluate \(\sum_{r=2}^{5} \frac{1}{r-1}\). [2]
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum_{r=2}^{a} f(r)\) where \(f(r)\) and \(a\) are to be determined. [2]
OCR MEI C2 2014 June Q2
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]
OCR MEI C2 Q3
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]