1.04g Sigma notation: for sums of series

1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 }\) is defined by

$$a _ { n } = \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right)$$ Find the exact values of
a. i) \(a _ { 1 }\) ii) \(a _ { 2 }\) iii) \(a _ { 3 }\) b. Hence find the exact value of $$\sum _ { n = 1 } ^ { 100 } \left\{ n + \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right) \right\}$$

1. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

1. (i) Show that \(\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798\) (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$u _ { n + 1 } = \frac { 1 } { u _ { n } } , \quad u _ { 1 } = \frac { 2 } { 3 }$$ Find the exact value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\)

1. (i) Find the value of

$$\sum _ { r = 4 } ^ { \infty } 20 \times \left( \frac { 1 } { 2 } \right) ^ { r }$$ (3)
(ii) Show that $$\sum _ { n = 1 } ^ { 48 } \log _ { 5 } \left( \frac { n + 2 } { n + 1 } \right) = 2$$

1. A sequence of terms \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = 8 - a _ { n } \end{aligned}$$

1. 1. Show that this sequence is periodic.
   2. State the order of this periodic sequence.
2. Find the value of $$\sum _ { n = 1 } ^ { 85 } a _ { n }$$

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by

$$\begin{aligned} u _ { 1 } & = 35 \\ u _ { n + 1 } & = u _ { n } + 7 \cos \left( \frac { n \pi } { 2 } \right) - 5 ( - 1 ) ^ { n } \end{aligned}$$

1. 1. Show that \(u _ { 2 } = 40\)
   2. Find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\) Given that the sequence is periodic with order 4
2. 1. write down the value of \(u _ { 5 }\)
   2. find the value of \(\sum _ { r = 1 } ^ { 25 } u _ { r }\)

4

1. The first four terms of a sequence are \(2,3,0,3\) and the subsequent terms are given by \(\mathrm { a } _ { \mathrm { k } + 4 } = \mathrm { a } _ { \mathrm { k } }\).
   1. State what type of sequence this is.
   2. Find \(\sum _ { \mathrm { k } = 1 } ^ { 200 } \mathrm { a } _ { \mathrm { k } }\).
2. A different sequence is given by \(\mathrm { u } _ { \mathrm { n } } = \mathrm { b } ^ { \mathrm { n } }\) where \(b\) is a constant and \(n \geqslant 1\).
   1. State the set of values of \(b\) for which this is a divergent sequence.
   2. In the case where \(b = \frac { 1 } { 3 }\), find the sum of all the terms in the sequence.

11

1. Evaluate \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).
2. Show that Euler's approximate formula, as given in line 13, gives the exact value of \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).

1 Find the value of \(\sum _ { r = 1 } ^ { 5 } 2 ^ { r } ( r - 1 )\).

6 In this question you must show detailed reasoning.
Show that \(\sum _ { r = 1 } ^ { 3 } \frac { 1 } { \sqrt { r + 1 } + \sqrt { r } } = 1\).

4 Show that \(\sum _ { r = 1 } ^ { 4 } \ln \frac { r } { r + 1 } = - \ln 5\).

3 An arithmetic series has fifth term 46 and twentieth term 181.

1. 1. Show that the common difference is 9 .
   2. Find the first term.
2. Find the sum of the first 20 terms of the series.
3. The \(n\)th term of the series is \(u _ { n }\). Given that the sum of the first 50 terms of the series is 11525 , find the value of $$\sum _ { n = 21 } ^ { 50 } u _ { n }$$

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 31 terms of the series is 310 .

1. Show that \(a + 15 d = 10\).
2. Given also that the 21st term is twice the 16th term, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).

6 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 25 terms of the series is 3500 .

1. Show that \(a + 12 d = 140\).
2. The fifth term of this series is 100 . Find the value of \(d\) and the value of \(a\).
3. The \(n\)th term of this series is \(u _ { n }\). Given that $$33 \left( \sum _ { n = 1 } ^ { 25 } u _ { n } - \sum _ { n = 1 } ^ { k } u _ { n } \right) = 67 \sum _ { n = 1 } ^ { k } u _ { n }$$ find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
   (3 marks)

8

1. The equation $$x ^ { 3 } + 2 x ^ { 2 } + x - 100000 = 0$$ has one real root. Taking \(x _ { 1 } = 50\) as a first approximation to this root, use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to the root.
2. 1. Given that \(S _ { n } = \sum _ { r = 1 } ^ { n } r ( 3 r + 1 )\), use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$S _ { n } = n ( n + 1 ) ^ { 2 }$$
   2. The lowest integer \(n\) for which \(S _ { n } > 100000\) is denoted by \(N\). Show that $$N ^ { 3 } + 2 N ^ { 2 } + N - 100000 > 0$$
3. Find the value of \(N\), justifying your answer.

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that

$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { 1 } { 2 } n \left[ 6 n ^ { 2 } - 3 n - 1 \right]$$ for all positive integers \(n\).
(b) Hence find any values of \(n\) for which $$\sum _ { r = 5 } ^ { n } ( 3 r - 2 ) ^ { 2 } + 103 \sum _ { r = 1 } ^ { 28 } r \cos \left( \frac { r \pi } { 2 } \right) = 3 n ^ { 3 }$$

5. \begin{figure}[h] \end{figure} A block has length \(( r + 2 ) \mathrm { cm }\), width \(( r + 1 ) \mathrm { cm }\) and height \(r \mathrm {~cm}\), as shown in Figure 2.
In a set of \(n\) such blocks, the first block has a height of 1 cm , the second block has a height of 2 cm , the third block has a height of 3 cm and so on.

1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that the total volume, \(V\), of all \(n\) blocks in the set is given by $$V = \frac { n } { 4 } ( n + 1 ) ( n + 2 ) ( n + 3 ) \quad n \geqslant 1$$ Given that the total volume of all \(n\) blocks is $$\left( n ^ { 4 } + 6 n ^ { 3 } - 11710 \right) \mathrm { cm } ^ { 3 }$$
2. determine how many blocks make up the set.

1 In this question you must show detailed reasoning. Find \(\sum _ { r = 2 } ^ { 50 } \left( \frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \right)\), expressing the answer as an exact fraction.

7 The diagram shows part of the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762} The area \(A\) of the region enclosed by the curve, the \(x\)-axis and the line \(x = p\) is given approximately by the sum \(S\) of the areas of \(n\) rectangles, each of width \(h\), where \(h\) is small and \(n h = p\). The first three such rectangles are shown in the diagram.

1. Find an expression for \(S\) in terms of \(n\) and \(h\).
2. Use the identity \(\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )\).
3. Show how to use this result to find \(A\) in terms of \(p\).

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

      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
   4. (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

8 Evaluate the following, giving your answers in exact form.

1. \(\sum _ { n = 1 } ^ { 30 } \frac { 1 } { n } - \sum _ { n = 2 } ^ { 29 } \frac { 1 } { n }\).
2. \(\sum _ { n = 1 } ^ { 100 } n \times ( - 1 ) ^ { n }\).

5 An arithmetic progression has first term 5 and common difference 7.

1. Find the value of the 10th term.
2. Find the sum of the first 15 terms. The terms of the progression are given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\).
3. Evaluate \(\sum _ { n = 1 } ^ { 15 } \left( 2 x _ { n } + 1 \right)\).

1 The series \(S\) is given by \(S = \sum _ { r = 0 } ^ { N } ( N + r ) ^ { 2 }\).

1. Write out the first three terms and the last three terms of the series for \(S\).
2. Use the standard result \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } N ( N + 1 ) ( a N + 1 )\) for some positive integer \(a\) to be determined.