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Find, in terms of π, the general term of the sequence cos two π, cos four π, cos six π, cos eight π, and so forth.
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In this question, weβre given a sequence and weβre asked to find the general term in terms of this letter π.
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And when weβre solving problems with sequences, π is usually taken to be an index.
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This index or π-value will indicate a position in the sequence.
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Usually, the first term of a sequence has an index of one, the second term has an index of two, and the third and fourth terms have an index π of three and four.
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Having a general term in terms of π will allow us to work out any term in the sequence.
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For example, if we wanted to calculate the 20th term, we would simply substitute π is equal to 20 into the general term.
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And so if we look at this sequence, we can identify that every term is the cosine of an angle.
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In fact, every term in this sequence is the cosine of some angle which is something multiplied by π.
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The value that changes in each term is the coefficient of π.
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And so we must ask, how does the index π in each term relate to the coefficient of π?
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Well, since each coefficient of π is double the index, then we can say that, for any value of π, the value in the sequence is cos of two π multiplied by π.
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We can therefore give the answer that the general term of this sequence is cos of two ππ.
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And if, for example, we did wish to work out the 20th term, we could go ahead and calculate that the 20th term would be cos of 40π.