1.04g Sigma notation: for sums of series

5 It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r }\), where \(u _ { r } = x ^ { \mathrm { f } ( r ) } - x ^ { \mathrm { f } ( r + 1 ) }\) and \(x > 0\).

1. Find \(S _ { n }\) in terms of \(n , x\) and the function f .
2. Given that \(\mathrm { f } ( r ) = \ln r\), find the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-10_2716_31_106_2016} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-11_2723_35_101_20}
3. Given instead that \(\mathrm { f } ( r ) = 2 \log _ { x } r\) where \(x \neq 1\), use standard results from the List of formulae (MF19) to find \(\sum _ { n = 1 } ^ { N } S _ { n }\) in terms of \(N\). Fully factorise your answer.

11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.

1. Show that \(p = 9\)
2. Find the value of the 20th term of this series.
3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}

1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies

$$u _ { n + 1 } = 2 u _ { n } - 6 , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 2\)

1. find the value of \(u _ { 3 }\)
2. evaluate \(\sum _ { i = 1 } ^ { 4 } u _ { i }\)

2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = 2 - 3 u _ { n } \quad n \geqslant 1 \end{aligned}$$

1. Find the value of \(u _ { 2 }\) and the value of \(u _ { 3 }\)
2. Calculate the value of \(\sum _ { r = 1 } ^ { 4 } \left( r - u _ { r } \right)\) □

4. A sequence is defined by $$\begin{aligned} u _ { 1 } & = k , \text { where } k \text { is a constant } \\ u _ { n + 1 } & = 4 u _ { n } - 3 , n \geqslant 1 \end{aligned}$$

1. Find \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\), simplifying your answers as appropriate. Given \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 18\)
2. find \(k\).

5. (i) $$U _ { n + 1 } = \frac { U _ { n } } { U _ { n } - 3 } , \quad n \geqslant 1$$ Given \(U _ { 1 } = 4\), find

1. \(U _ { 2 }\)
2. \(\sum _ { n = 1 } ^ { 100 } U _ { n }\) (ii) Given $$\sum _ { r = 1 } ^ { n } ( 100 - 3 r ) < 0$$ find the least value of the positive integer \(n\).

7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 100 }\)
3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)

4. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n } = k n - 3 ^ { n }$$ where \(k\) is a constant. Given that \(u _ { 2 } = u _ { 4 }\)

1. find the value of \(k\)
2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\)

7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { 1 } = 2 \\ a _ { n + 1 } = 3 a _ { n } - c \end{gathered}$$ where \(c\) is a constant.

1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
2. find the value of \(c\).

5. The \(r\) th term of an arithmetic series is ( \(2 r - 5\) ).

1. Write down the first three terms of this series.
2. State the value of the common difference.
3. Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).

4. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n + 1 } = 2 u _ { n } - 1 , n \geqslant 1$$ Given that \(u _ { 2 } = 9\),

1. find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\),
2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\).

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 \(a _ { 6 }\)

4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 3 a _ { n } - 5 , \quad n \geqslant 1 . \end{aligned}$$

1. Find the value of \(a _ { 2 }\) and the value of \(a _ { 3 }\).
2. Calculate the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\).

8. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 3 a _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive integer.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 9 k + 20\).
3. 1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\).
   2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 10 .

7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 2 a _ { n } - 7 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 4 k - 21\). Given that \(\sum _ { r = 1 } ^ { 4 } a _ { r } = 43\),
3. find the value of \(k\).

5. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 5 a _ { n } + 3 , \quad n \geqslant 1 , \end{aligned}$$ where \(k\) is a positive integer.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 25 k + 18\).
3. 1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\), in its simplest form.
   2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 6 .

6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 \\ x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1 \end{gathered}$$ where \(k\) is a constant, \(k \neq 0\)

1. Find an expression for \(x _ { 2 }\) in terms of \(k\).
2. Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\) Given also that \(x _ { 3 } = 1\),
3. calculate the value of \(k\).
4. Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)

6. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = 5 - k a _ { n } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\). Find
2. \(\sum _ { r = 1 } ^ { 3 } \left( 1 + a _ { r } \right)\) in terms of \(k\), giving your answer in its simplest form,
3. \(\sum _ { r = 1 } ^ { 100 } \left( a _ { r + 1 } + k a _ { r } \right)\)

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

$$\begin{aligned} a _ { 1 } & = 1 \\ a _ { n + 1 } & = \frac { k \left( a _ { n } + 1 \right) } { a _ { n } } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive constant.

1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\), giving your answers in their simplest form. Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 10\)
2. find an exact value for \(k\).

8. (i) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum to \(n\) terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of

1. \(u _ { 5 }\)
2. \(\sum _ { n = 1 } ^ { 59 } u _ { n }\)

1. (i) Using the laws of logarithms, solve

$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$ (ii) Given that $$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } y \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$ find \(y\) in terms of \(a\), giving your answer in simplest form.

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

$$\begin{aligned} a _ { n + 1 } & = 4 - a _ { n } \\ a _ { 1 } & = 3 \end{aligned}$$ Find the value of

1. 1. \(a _ { 2 }\)
   2. \(a _ { 107 }\)
2. \(\sum _ { n = 1 } ^ { 200 } \left( 2 a _ { n } - 1 \right)\)

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

$$a _ { n } = \cos ^ { 2 } \left( \frac { \mathrm { n } \pi } { 3 } \right)$$ Find the exact values of

1. 1. \(a _ { 1 }\)
   2. \(a _ { 2 }\)
   3. \(a _ { 3 }\)
2. Hence find the exact value of 50 $$n + \cos ^ { 2 } \frac { n \pi } { 3 }$$ You must make your method clear.

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

$$\begin{gathered} u _ { 1 } = 3 \\ u _ { n + 1 } = 2 - \frac { 4 } { u _ { n } } \end{gathered}$$

1. Find the value of \(u _ { 2 }\), the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\)
2. Find the value of $$\sum _ { r = 1 } ^ { 100 } u _ { r }$$