Find the next four terms in a geometric sequence 29, negative 58 thirds, 116 ninths.
Geometric sequences are sequences that have a common ratio. A ratio that you multiply by one term to get the next term. To find the next four terms, we’ll need to find out what the common ratio is for this sequence. This is the formula we’ll use to find our common ratio. To use this formula, first we’ll need to choose a numerator, a term to start with. Let’s start with negative 58 thirds.
The way this formula works and the way that we need to choose our denominator is that whatever term we select for the numerator, the next term that we choose for the denominator must be one before it. Since we started with negative 58 thirds, our next term down, our previous term would be what? 29. We plug in 29 for our denominator because 29 is the previous term from the term we’re using for the numerator. So we choose the term and we choose the previous term.
Let’s rewrite our division problem like this. And since we can’t divide a fraction by a fraction, we’ll multiply 58 thirds, negative 58 thirds by one over 29. When I multiply those things together, I get negative 58 over 87 which reduces to negative two-thirds, which reduces to negative two-thirds. This tells us that our common ratio, what is being multiplied by a term to produce the next term in line, is negative two-thirds.
Our question is asking for the next four terms in the sequence. So we take our third term in the sequence, 116 over nine, and we multiply that by negative two-thirds. This gives us negative 232 over 27. We take the fourth term, bring it down, and multiply 232 over 27 by negative two-thirds, our common ratio, which gives us 464 over 81. We bring down our fifth term, multiply 464 over 81 times negative two-thirds. Our next term, negative 928 over 243.
Running out of space here, so I’m gonna drag this negative 928 over 243 to the right, and multiply that by negative two-thirds which would give us 1856 over 729.
This would be the next four terms of the geometric sequence with the common ratio negative two-thirds.