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, βˆ’5, 5/2, βˆ’5/4, ….

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‘Ž 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 I’m just gonna label all of my terms π‘Ž one, π‘Ž two, π‘Ž three, and π‘Ž four, and so on. And then I’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‘Ÿ, I get the second term. If I multiply the second term by the common ratio, π‘Ÿ, I get the third term. If I multiply the third term by the common ratio, π‘Ÿ, 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‘Ž one, I get that π‘Ÿ is equal to π‘Ž two over π‘Ž one. Now, if I divide both sides of the second equation by π‘Ž two, I get that π‘Ÿ is equal to π‘Ž three over π‘Ž two and similarly for the third equation.

So to work out the value of π‘Ÿ, I just take the value of one term and divide it by the value of the previous term. Now remember in a geometric sequence, it’s a common ratio. So it doesn’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‘Ÿ. Well looking at these numbers here, the easiest pair to use is gonna be π‘Ž one and π‘Ž two. So π‘Ÿ is equal to π‘Ž two divided by π‘Ž one. π‘Ž two is negative five and π‘Ž 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‘Ž one equals 10 and π‘Ÿ equals negative a half, uniquely defines this sequence. When we know this, we can generate all the terms in the sequence if we’re prepared to put in enough time and doing enough multiplying.

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