Question Video: Identifying Whether a given Sequence Is Increasing, Decreasing, or Neither Mathematics

Is the sequence π‘Ž_𝑛 = ((βˆ’1)^𝑛)/(11𝑛) βˆ’ 22 increasing, decreasing, or neither?


Video Transcript

Is the sequence π‘Ž sub 𝑛 equals negative one to the power of 𝑛 over 11𝑛 minus 22 increasing, decreasing, or neither?

When we’re considering if a sequence is increasing or decreasing, we’re comparing any term to the term before it. If a sequence is increasing, then any term π‘Ž sub 𝑛 must be greater than π‘Ž sub 𝑛 minus one. That must be true for all values of 𝑛. Similarly, if a sequence is decreasing, then any term of index 𝑛 in a sequence must be less than the term before it. What we can do is to work out the first few terms of the sequence and see if the values are increasing, decreasing, or neither.

So we could take the 𝑛th term, and we’ll start by substituting 𝑛 is equal to one. So for the first term π‘Ž sub one, we have negative one to the power of one over 11 times one minus 22. When we simplify this, we get the fraction negative 243 over 11. Now that we’ve found the first term, we can find the second term by substituting in 𝑛 is equal to two. When we simplify negative one squared over 11 times two minus 22, we get the fraction negative 483 over 22. We can find the third term in the same way by substituting 𝑛 is equal to three. This means that we get a third term, π‘Ž sub three, equal to negative 727 over 33.

We have at this point got three terms in the sequence, but it’s a little difficult to see if they’re increasing or decreasing. So it might be helpful to find their decimal equivalents. The first term is approximately negative 22.09, the second term approximately negative 21.95, and the third term is approximately negative 22.03. We notice that the second term is greater than the first term. However, the third term is less than the second term. That means that we can’t say that for all values either π‘Ž sub 𝑛 is bigger than π‘Ž sub 𝑛 minus one or π‘Ž sub 𝑛 is less than π‘Ž sub 𝑛 minus one. And that means that the sequence isn’t increasing or decreasing, so it must be neither. The answer is that π‘Ž sub 𝑛 is neither increasing nor decreasing.

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