# Video: Finding the First Term and the Common Ratio in a Geometric Sequence

Write down the first term and the common ratio for the following geometric sequence: 10, β5, 5/2, β5/4, β¦.

02:09

### Video Transcript

Write down the first term and the common ratio for the following geometric sequence: 10, negative five, five over two, negative five over four, and so on.

Well clearly, the first term is equal to 10. So π one is equal to 10; that bit was quite easy. And the common ratio is what do we multiply each term by to get the next term. So Iβm just gonna label all of my terms π one, π two, π three, and π four, and so on. And then Iβm just going to write a little formula for how do I get from one term to the next term. Well, if I multiply the first term by the common ratio, π, I get the second term. If I multiply the second term by the common ratio, π, I get the third term. If I multiply the third term by the common ratio, π, I get the fourth term, and so on and so on and so on. So looking at that first equation, if I divide both sides of the equation by π one, I get that π is equal to π two over π one. Now, if I divide both sides of the second equation by π two, I get that π is equal to π three over π two and similarly for the third equation.

So to work out the value of π, I just take the value of one term and divide it by the value of the previous term. Now remember in a geometric sequence, itβs a common ratio. So it doesnβt matter whether I take the second and the first or the third and the second or the fourth and the third. As long as I take two consecutive terms, I will always find the same answer for π. Well looking at these numbers here, the easiest pair to use is gonna be π one and π two. So π is equal to π two divided by π one. π two is negative five and π one is ten. So the common ratio is negative five divided by 10 and that simplifies to negative a half.

So to get from each term to the next term, I have to multiply by negative a half. 10 times negative a half is minus five, negative five times negative a half is five over two, and so on. So these two facts here, π one equals 10 and π equals negative a half, uniquely defines this sequence. When we know this, we can generate all the terms in the sequence if weβre prepared to put in enough time and doing enough multiplying.