Video: Finding the Terms of a Sequence Given Its General Term

Which of the following are the first five terms of the sequence with general term 𝑎_(𝑛) = −99 − 17/√(𝑛) and 𝑛 ≥ 1? [A] −82, −181/2, −280/3, −379/4, −478/5, ... [B] −116, −215/2, −314/3, −413/4, −512/5, ... [C] −116, −99 + (17√2)/2, −99 + (17√3)/3, −215/2, −99 + (17√5)/5, ... [D] −116, −99 − (17√2)/2, −99 − (17√3)/3, −215/2, −99 − (17√5)/5, ...

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Video Transcript

Which of the following are the first five terms of the sequence with general term 𝑎 sub 𝑛 equals negative 99 minus 17 over the square root of 𝑛 and 𝑛 must be greater than or equal to one?

We’re given the general term for a sequence. We’re told that 𝑎 sub 𝑛 equals negative 99 minus 17 over the square root of 𝑛 and that 𝑛 must be greater than or equal to one. When we’re dealing with sequences, the value of 𝑛 represents the term number. And that means when 𝑛 equals one, we’re calculating the first term in the sequence.

𝑎 sub one is equal to negative 99 minus 17 over the square root of one. The square root of one is one. 17 over one equals 17. And negative 99 minus 17 equals negative 116. The first term then must be negative 116. And we can eliminate option A.

To continue with this process, we’ll find 𝑎 sub two, which is the second term. We plug in two for our 𝑛-value and we get negative 99 minus 17 over the square root of two. If we want to rationalise and get the square root of two out of the denominator, we can multiply the fraction 17 over the square root of two by the square root of two over two. The negative 99 doesn’t change. 17 times the square root of two is written like this. But the square root of two times the square root of two equals two. We no longer have the square root in the denominator.

Our second term is then negative 99 minus 17 times the square root of two over two, which eliminates option B. And if we look closely, it eliminates option C as well. Option C says the second term is negative 99 plus 17 times the square root of two over two. But we know that it is minus. And we see that in option D.

And so we say that the first five terms of this sequence will be negative 116. Negative 99 minus 17 times the square root of two over two. Negative 99 minus 17 times the square root of three over three. Negative 215 over two. Negative 99 minus 17 times the square root of five over five. And it continues. The key here was recognising that the 𝑛-value represents the term number and plugging in the terms you were looking for.