Question Video: Finding the Value of a Term in a Geometric Sequence Mathematics

Video Transcript

The first term of a geometric sequence is two, and its third term is seven. If the common ratio is negative, what is the second term?

A geometric sequence is one in which there’s a common ratio between terms. To get from one term to the next, we always multiply by the same value, which we call the common ratio and denote using the letter 𝑟. We’re told that the first term in this geometric sequence is two and the third term is seven. The value of the second term is what we need to work out. Well, to get from the first term to the second, we multiply by 𝑟. And then, to get from the second term to the third term, we multiply by 𝑟 again. Overall then, we’ve multiplied by 𝑟 squared, and so we can form an equation. Two 𝑟 squared is equal to seven. We can solve this equation to determine the value of the common ratio.

Dividing both sides by two, we have that 𝑟 squared is equal to seven over two. We then take the square root of each side of this equation, giving 𝑟 equals plus or minus the square root of seven over two. If we look carefully at the question, though, we’re told that the common ratio is negative. So the value of 𝑟 for this question is negative the square root of seven over two. To find the second term in the sequence then, we need to take the first term, which was two, and multiply it by this common ratio. This gives two multiplied by negative the square root of seven over two.

Using laws of surds or radicals, we can say that the square root of seven over two is the square root of seven over the square root of two. So this becomes two multiplied by negative root seven over root two. We then see that we have two divided by the square root of two. And any positive number divided by its own square root will give its square root. Two divided by the square root of two is equal to the square root of two. So this becomes the square root of two multiplied by negative the square root of seven.

We can then apply one further rule of surds or radicals. The square root of 𝑎 multiplied by the square root of 𝑏 is the square root of 𝑎𝑏. So the square root of two multiplied by the square root of seven is the square root of 14. And this is all multiplied by negative one. So we find that the second term in this geometric sequence is negative the square root of 14.

We can check our answer because if we multiply by the common ratio again, we should get the third term, which is seven. So we multiply negative root 14 by the common ratio of negative the square root of seven over two. The two negatives form a positive. And then using the rule we wrote down for radicals earlier on, this is equal to the square root of 14 multiplied by seven over two. We can then cancel a factor of two from the numerator and denominator, giving the square root of seven multiplied by seven. That’s the square root of seven squared or the square root of 49, which is equal to seven. And so this confirms that our answer is correct. If the common ratio is negative, the second term in this geometric sequence is negative the square root of 14.