Video Transcript
Find the second and third terms of a geometric sequence given that the first term is 69, the seventh term is equal to 4416, and all terms are positive.
So, the first bit of important information we’re told is that we’re looking at a geometric sequence. And a geometric sequence is a sequence where there is a common ratio between each term. And when we’re dealing with geometric sequences, what we have is a general form for each one of the terms. And that is that 𝑎 sub 𝑛 is equal to 𝑎 sub one multiplied by 𝑟 to the power of 𝑛 minus one, where we’ve got 𝑎 sub 𝑛 is the term, we’ve got 𝑎 sub one is our first term, we’ve got 𝑟 is our common ratio, and then we have 𝑛 is our term number. Okay, great. So, we’ve got this. So, how can we use it now to find out what the second and third terms of our sequence are?
Well, we know that the first term is 69 and the seventh term is 4416. Well, if we apply our general form, we can say that the seventh term is equal to the first term multiplied by 𝑟 to the power of seven minus one. So therefore, what we can say is that 4416 is equal to 69 multiplied by 𝑟 to the power of six. And we got that cause we’ve substituted in the values we know. So then, what we can do is divide each side of the equation by 69. And when we do that, we’re gonna get 64 is equal to 𝑟 to the power of six. So then, what we’re gonna do is take the sixth root of both sides of the equation. And when we do that, we’re gonna get two is equal 𝑟. So, we now know that our common ratio is equal to two.
So now, as we have the first term and the common ratio, we can use these to find the second and third terms of our geometric sequence. So, first of all, for the second term, we’re gonna have 𝑎 sub two is equal to 69 multiplied by two to the power of one. We don’t usually write one, but I’ve just written it there so we can see what’s happening. So therefore, we’re gonna get that the second term is equal to 138. And then for the third term, we’re gonna have 69 multiplied by two squared, which is gonna give us 69 multiplied by four, which is equal to 276. So therefore, we can say that the second and third terms are 138 and 276, respectively.
