Write the first four terms of the sequence that begins with 5000 and decays 15% with each term. What is the common ratio?
I found that that common ratio is .85 but I'm confused on how yyou would get that?
What is .85? it is 1-.15
To find the common ratio of the given sequence, we can multiply each term by the same value to obtain the next term. In this case, the sequence begins with 5000, and each subsequent term is obtained by decaying 15% of the previous term.
To decay a value by 15%, we multiply it by (100% - 15%), which is 85% or 0.85 in decimal form.
Let's find the first four terms of the sequence:
1st term: 5000
2nd term: 5000 * 0.85 = 4250
3rd term: 4250 * 0.85 = 3612.50
4th term: 3612.50 * 0.85 = 3070.63
So, the first four terms of the sequence are 5000, 4250, 3612.50, and 3070.63. And as you correctly mentioned, the common ratio is 0.85.
To find the common ratio of a geometric sequence, you need to analyze the pattern of the terms. In this case, you're given that the sequence begins with 5000 and decays 15% with each term.
Let's break down the steps to find the common ratio:
1. Start with the initial term: The first term of the sequence is 5000.
2. Compute the second term: To find the second term, you need to decrease the first term by 15%. Subtract 15% of 5000 from 5000:
Second Term = 5000 - (0.15 * 5000) = 5000 - 750 = 4250
3. Compute the third term: Similar to the previous step, you need to decrease the second term by 15%.
Third Term = 4250 - (0.15 * 4250) = 4250 - 637.5 = 3612.5
4. Compute the fourth term: Again, decrease the third term by 15%.
Fourth Term = 3612.5 - (0.15 * 3612.5) = 3612.5 - 541.875 = 3070.625
Now let's examine the pattern:
From the initial term (5000) to the second term (4250), we can see that the second term is 85% of the first term.
From the second term (4250) to the third term (3612.5), we can observe that the third term is also 85% of the second term.
Similarly, the fourth term (3070.625) is 85% of the third term (3612.5).
This pattern shows that each term is obtained by multiplying the previous term by 0.85, or equivalently, each term is 85% of the previous term.
Therefore, the common ratio is 0.85.