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