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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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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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 ,
Show that
For this value of , calculate .
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A geometric series is given by
Write down the common ratio, , of the series.
Given that the series is convergent, and that , calculate the value of .
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An arithmetic series is given by
Given that and , find the values of and .
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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
Calculate the value of .
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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 ,
(i) show that
(ii) hence find the value of .
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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Prove that, for all ,
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