Sequence defined by formula

7

1. In an arithmetic progression, the sum of the first \(n\) terms, denoted by \(S _ { n }\), is given by $$S _ { n } = n ^ { 2 } + 8 n .$$ Find the first term and the common difference.
2. In a geometric progression, the second term is 9 less than the first term. The sum of the second and third terms is 30 . Given that all the terms of the progression are positive, find the first term.

3

1. Each year, the value of a certain rare stamp increases by \(5 \%\) of its value at the beginning of the year. A collector bought the stamp for \(\\) 10000\( at the beginning of 2005. Find its value at the beginning of 2015 correct to the nearest \)\\( 100\).
2. The sum of the first \(n\) terms of an arithmetic progression is \(\frac { 1 } { 2 } n ( 3 n + 7 )\). Find the 1 st term and the common difference of the progression.

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 \(\sum _ { r = 1 } ^ { 4 } u _ { r }\)

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 \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).

9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 □ Row 2 □ 1 Row 3 \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-11_40_104_566_479} She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
2. Find the total number of sticks Ann uses in making these 10 rows. Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the ( \(k + 1\) )th row,
3. show that \(k\) satisfies \(( 3 k - 100 ) ( k + 35 ) < 0\).
4. Find the value of \(k\).

6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence it is.
3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).

1 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by $$u _ { n } = 3 n - 1 ,$$ for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\), and state what type of sequence \(S\) is.
2. Evaluate \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).

5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)

1. Write down the first three terms of the sequence.
2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).

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

$$u _ { n } = 2 ^ { n } + k n ,$$ where \(k\) is a constant.
Given that \(u _ { 1 } = u _ { 3 }\),

1. find the value of \(k\),
2. find the value of \(u _ { 5 }\).

2 The \(n\)th term of a sequence, \(u _ { n }\), is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\). State what type of sequence this is.
2. Find \(\sum _ { n = 1 } ^ { 30 } u _ { n }\).

3 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 24 - \frac { 2 } { 3 } n$$

1. Write down the exact values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(k\) such that \(u _ { k } = 0\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).

2 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence \(S\) is.
3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).

6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
4. Find the sum to infinity of the geometric progression in part (iii).

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 n + 1\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Evaluate \(\sum _ { n = 1 } ^ { 40 } u _ { n }\). Another sequence \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { 1 } = 2\) and \(w _ { n + 1 } = 5 w _ { n } + 1\).
3. Find the value of \(p\) such that \(u _ { p } = w _ { 3 }\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 3 n - 1\), for \(n \geqslant 1\).

1. Find the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 ^ { n } + k n ,$$ where \(k\) is a constant. Given that \(u _ { 1 } = u _ { 3 }\),

1. find the value of \(k\),
2. find the value of \(u _ { 5 }\).

3 The \(n\)th term of an arithmetic sequence is \(u _ { n }\), where $$u _ { n } = 90 - 3 n$$

1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
2. Write down the common difference of the arithmetic sequence.
3. Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).