Question Video: Finding the Geometric Means of a Given Geometric Sequence Mathematics

Find the geometric means of the sequence (2, ..., ..., ..., 4, 802).

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Video Transcript

Find the geometric means of the sequence whose first term is two and whose last term is 4802.

Remember, 𝑛 geometric means between two numbers are the 𝑛 terms of a geometric sequence between the two given numbers. So, we’re going to need to identify the sequence whose first term is two and whose last term is 4802 and where there are exactly three terms between them. So, we use the formula for the 𝑛th term of a geometric sequence with first term π‘Ž sub one and common ratio π‘Ÿ. It’s π‘Ž sub 𝑛 equals π‘Ž sub one times π‘Ÿ to the power of 𝑛 minus one.

The first term in our sequence is two; then the fifth term is 4802. But then using our formula, we can express that in terms of π‘Ÿ; it’s two times π‘Ÿ to the power of five minus one or two times π‘Ÿ to the fourth power. So, we now have an equation which we can solve to find the value of π‘Ÿ.

We begin by dividing both sides of this equation by two. So, π‘Ÿ to the fourth power is 2401. Then, we can find the positive and negative fourth root of 2401. And remember, we do this because four is an even exponent or power, and this gives us a value for π‘Ÿ as positive or negative seven. And so, in fact, there are two possible sequences we’re interested in. The first is when π‘Ÿ is equal to seven. Beginning with our first term two, we multiply by seven each time. And this gives us the sequence two, 14, 98, 686, and 4802. And if π‘Ÿ is negative seven, we change the sign of the 14 and the 686. So, the geometric means of this sequence are either 14, 98, 686 or negative 14, 98, and negative 686.

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