1.04e Sequences: nth term and recurrence relations

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

1. Find an expression for \(x _ { 2 }\) in terms of \(a\).
2. Show that \(x _ { 3 } = a ^ { 2 } - 3 a - 3\). Given that \(x _ { 3 } = 7\),
3. find the possible values of \(a\).

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 \(k\).

1. A sequence of positive numbers is defined by

$$\begin{aligned} a _ { n + 1 } & = \sqrt { } \left( a _ { n } ^ { 2 } + 3 \right) , \quad n \geqslant 1 , \\ a _ { 1 } & = 2 \end{aligned}$$

1. Find \(a _ { 2 }\) and \(a _ { 3 }\), leaving your answers in surd form.
2. Show that \(a _ { 5 } = 4\)

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 .

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

1. Write down an expression, in terms of \(c\), for \(a _ { 2 }\)
2. Show that \(a _ { 3 } = 12 - 3 c\) Given that \(\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23\)
3. find the range of values of \(c\).

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 \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)

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

1. Find an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 2\),
2. find the two possible values of \(k\).

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 \(k\).

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

$$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = \frac { a _ { n } } { a _ { n } + 1 } , \quad n \geqslant 1 , n \in \mathbb { N } \end{aligned}$$

1. Find the values of \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\) Write your answers as simplified fractions. Given that $$a _ { n } = \frac { 4 } { p n + q } , \text { where } p \text { and } q \text { are constants }$$
2. state the value of \(p\) and the value of \(q\).
3. Hence calculate the value of \(N\) such that \(a _ { N } = \frac { 4 } { 321 }\)

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

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

1. Find an expression for \(a _ { 2 }\) in terms of \(p\), giving your answer in simplest form. Given that \(\sum _ { n = 1 } ^ { 3 } a _ { n } = p + 15\)
2. find the possible values of \(a _ { 2 }\)

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1. 1. Find the value of

$$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$ (3)
(ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N } \end{aligned}$$

1. Show that this sequence is periodic.
2. State the order of this sequence.
3. Hence find $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$

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

$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n } \\ u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.

1. Find, in terms of \(a\) and \(b\),
   1. \(u _ { 2 }\)
   2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
2. \(b = a + 9\)
3. show that
$$a ^ { 2 } - 5 a - 66 = 0$$
4. Hence find the larger possible value of \(u _ { 2 }\)

1. The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 16 }\) are

$$1 , - 4 x \text { and } p x ^ { 2 }$$ where \(k\) and \(p\) are constants.

1. Find, in simplest form,
   1. the value of \(k\)
   2. the value of \(p\) $$g ( x ) = \left( 2 + \frac { 16 } { x } \right) ( 1 + k x ) ^ { 16 }$$ Using the value of \(k\) found in part (a),
2. find the term in \(x ^ { 2 }\) in the expansion of \(\mathrm { g } ( x )\). $$\begin{aligned} u _ { 1 } & = 6 \\ u _ { n + 1 } & = k u _ { n } + 3 \end{aligned}$$ where \(k\) is a positive constant.
3. Find, in terms of \(k\), an expression for \(u _ { 3 }\) Given that \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 117\)
4. find the value of \(k\).

2. A sequence is defined by

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 $$\sum _ { r = 1 } ^ { 100 } u _ { r }$$

5. The first three terms of a geometric series are \(4 p , ( 3 p + 15 )\) and ( \(5 p + 20\) ) respectively, where \(p\) is a positive constant.

1. Show that \(11 p ^ { 2 } - 10 p - 225 = 0\)
2. Hence show that \(p = 5\)
3. Find the common ratio of this series.
4. Find the sum of the first ten terms of the series, giving your answer to the nearest integer.

1. The first three terms of a geometric series are

$$18,12 \text { and } p$$ respectively, where \(p\) is a constant. Find

1. the value of the common ratio of the series,
2. the value of \(p\),
3. the sum of the first 15 terms of the series, giving your answer to 3 decimal places.

9. The first three terms of a geometric sequence are $$7 k - 5,5 k - 7,2 k + 10$$ where \(k\) is a constant.

1. Show that \(11 k ^ { 2 } - 130 k + 99 = 0\) Given that \(k\) is not an integer,
2. show that \(k = \frac { 9 } { 11 }\) For this value of \(k\),
3. 1. evaluate the fourth term of the sequence, giving your answer as an exact fraction,
   2. evaluate the sum of the first ten terms of the sequence.

6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 , \text { for } n \geqslant 1 .$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geqslant 1\).

6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 \text {, for } n \geqslant 1 \text {. }$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geqslant 1\).

3. A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 2 \\ u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 . \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1\).