Calculate
The sum given in part (a) is an arithmetic series.
Write down the first term and the common difference.
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Calculate
The sum given in part (a) is an arithmetic series.
Write down the first term and the common difference.
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Calculate
The sum given in part (a) is a geometric series.
Write down the first term and the common ratio.
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It is given that
where is a positive integer.
Determine if the series is arithmetic or geometric, justifying your answer.
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A Fibonacci sequence can be expressed as the following recurrence relation
Write down the first six terms of the Fibonacci sequence with .
Find
with
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A sequence is defined by the recurrence relation .
Write down the first five terms of the sequence.
Determine if the sequence is arithmetic or geometric, justifying your answer.
Find
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The term of an arithmetic series is given by .
Write the sum of the series, up to the term, in sigma notation.
The term of a geometric series is given by .
Write the sum of the series, up to the term, in sigma notation.
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Given that
determine the value of .
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A sequence is defined for by the recurrence relation with
Calculate
What value of would make every term of the sequence equal?
Find the range of values for that would ensure every term of the sequence is positive?
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The first terms of a series are given by .
Show that this is an arithmetic series, and determine its first term and common difference.
Given that ,
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The first terms of a series are given by .
Show that this is a geometric series, and determine its first term and common ratio.
Given that = 20470,
Show that
For this value of , calculate .
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A geometric series is given by
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An arithmetic series is given by
Given that and , find the values of and .
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A sequence is defined for by the recurrence relation
Calculate
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A sequence is defined for by the recurrence relation
Calculate, giving your answers as exact values
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A sequence is defined for by the recurrence relation
where is a constant.
Write down expressions for and in terms of .
Given that the sequence is periodic with order 2, and given as well that ,
Find the value of .
For the value of found in part (b)
Calculate
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The terms of a sequence are defined by for all .
State, with a reason, whether this sequence is increasing, decreasing, or neither.
It can be shown that, for all ,
Using that formula,
Calculate
Find the value of ,i.e. the sum of the squares of all the integers between 51 and 100 inclusive.
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Given that ,
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Given that and , find the values of and
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Given that ,
Show that
For this value of , calculate .
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A convergent geometric series is given by
Write down the range of possible values of .
Given that =24
Calculate the value of .
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A sequence is defined for by the recurrence relation
Calculate
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A sequence is defined for by the recurrence relation
Calculate, giving your answers as exact values
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A sequence is defined for by the recurrence relation
where is a constant.
Given that the sequence is periodic with order 2, and given as well that ,
Find the value of .
For the value of found in part (a),
Calculate
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The terms of a sequence are defined, for all by .
State, with a reason, whether this sequence is increasing, decreasing, or neither.
It can be shown that, for all ,
= and
Using those formulas,
Show that .
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Given that , find the value of .
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Given that ,
For this value of , calculate .
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Given that , and , find the values of and .
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A convergent geometric series is given by , where in all cases the square root symbol indicates the positive square root of the number in question.
Write down the range of possible values of .
Given that
Calculate the value of .
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A sequence is defined for by .
Calculate , giving your answer as an exact value.
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A sequence is defined for all by
Determine, giving reasons for your answer, whether the sequence is increasing, decreasing, or neither.
A different sequence is defined for all by
where is a real constant.
Given that the sequence is not periodic,
suggest a possible value for , giving a reason for your answer.
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A sequence is defined for by the recurrence relation
where and are real numbers.
Show that the sequence is periodic, and determine its order.
Given that and
determine the possible values of and .
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Prove that, for all ,
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