Sigma notation evaluation

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 \(a\).

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 \(a _ { 6 }\)

2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).

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.

7. (a) Evaluate $$\sum _ { r = 10 } ^ { 30 } ( 7 + 2 r )$$ (b) (i) Write down the formula for the sum of the first \(n\) positive integers.
(ii) Using this formula, find the sum of the integers from 100 to 200 inclusive.
(iii) Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3 .

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.

3

1. Find \(\sum _ { r = 1 } ^ { 5 } \frac { 21 } { r + 2 }\).
2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = a , \text { where } a \text { is an unknown constant, } \\ u _ { n + 1 } & = u _ { n } + 5 . \end{aligned}$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence.

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.

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.

11 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).

1 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).

5 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).

3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).

1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).

1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).

1 Find \(\int 7 x ^ { \frac { 5 } { 2 } } \mathrm {~d} x\).

1. Find \(\sum _ { r = 1 } ^ { 5 } \frac { 21 } { r + 2 }\).
2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = a , \text { where } a \text { is an unknown constant, } \\ u _ { n + 1 } & = u _ { n } + 5 . \end{aligned}$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence.

4. (a) Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\) (b) A sequence has \(n\)th term \(u _ { n } = \sin \left( 90 n ^ { \circ } \right) n \geq 1\)

1. Find the order of the sequence.
2. Find \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)

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

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

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.

1. Evaluate

$$\sum _ { r = 1 } ^ { 30 } ( 3 r + 4 ) .$$

1. (a) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (b) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.

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