Video Transcript
State whether the following is true or false: A geometric sequence is decreasing if its common ratio 𝑟 is an element of the open interval negative one to zero.
Remember, a geometric sequence is one that’s generated by multiplying each term by some nonzero common ratio, where the 𝑛th term is 𝑎 times 𝑟 to the power of 𝑛 minus one, where 𝑎 is the first term and 𝑟 is the common ratio. The sequence is said to be decreasing if for all values of 𝑛 𝑎 sub 𝑛 plus one is less than 𝑎 sub 𝑛. In other words, any term in the sequence should be smaller than the term that precedes it.
So, let’s think about a geometric sequence with a general first term 𝑎 and a common ratio which is between negative one and zero. If the first term is 𝑎, the second term is 𝑎𝑟 to the power of two minus one, which is simply 𝑎𝑟. Now, if 𝑎 is a positive number and then we multiply this by the common ratio which is between negative one and zero — in other words, it’s a very small negative number — we get 𝑎𝑟 as being negative. So, if the first term is positive, the second term is negative, meaning 𝑎 two is less than 𝑎 one.
But what happens if 𝑎 is a negative number, if the first term in our sequence is negative? Well, if this is the case, we’re multiplying a negative by a second negative number, so 𝑎𝑟 itself must actually be positive. In this case, 𝑎 sub two must be greater than 𝑎 sub one, and so the sequence cannot be decreasing. In fact, let’s demonstrate what happens even if we do have a positive first term to the third term in our sequence. If the first term is positive, 𝑎𝑟 squared is the product of a positive number and a squared negative. So, it’s positive. This means that the third term has to be greater than the second term because the second term is negative.
So, in fact, what’s going to happen no matter the starting value of our sequence is if the common ratio is between negative one and zero, the sign of our terms is going to alternate. The actual magnitude of each term because we’re multiplying by a number between negative one and zero is going to get smaller. So, the sign will change and the number itself will get closer and closer to zero.
The answer to this question then is false. A geometric sequence is not decreasing if its common ratio 𝑟 is in the open interval negative one to zero.
