Sigma notation evaluation

6
6
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8 \(\quad \mathrm { P } ( n ) = \sum _ { k = 0 } ^ { n } k ^ { 3 } - \sum _ { k = 0 } ^ { n - 1 } k ^ { 3 }\) where \(n\) is a positive integer.
8

1. Find \(\mathrm { P } ( 3 )\) and \(\mathrm { P } ( 10 )\) 8
2. Solve the equation \(\mathrm { P } ( n ) = 1.25 \times 10 ^ { 8 }\)

10

1. An arithmetic series is given by $$\sum _ { r = 5 } ^ { 20 } ( 4 r + 1 )$$ 10
2. 1. Write down the first term of the series.
      10
3. (ii) Write down the common difference of the series.
   10
4. (iii) Find the number of terms of the series.
   

   10

   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]

3 A sequence is defined by $$u _ { 1 } = a \text { and } u _ { n + 1 } = - 1 \times u _ { n }$$ Find \(\sum _ { n = 1 } ^ { 95 } u _ { n }\) Circle your answer. \(- a\) 0 \(a\) 95a

1 Find $$\sum _ { r = 1 } ^ { 20 } \left( r ^ { 2 } - 2 r \right)$$ Circle your answer. 24502660532043680