Terms 1 through 5 of a sequence are shown below.
8/81 , 4/27 , 2/9 , 1/3 , 1/2
What is the 10th term in the sequence?
I have no clue. I tried having common denominator of 81, but that just confused me more. I would try to show more work, but I really don't know how to approach the problem?
Find the ratio of the 1st two terms:
(4/27) / (8/81) = 4/27 * 81/8 = 3/2
You can easily verify that each term is 3/2 the previous one. So,
So, a10 = 8/81 * (3/2)^9
= 2^3/3^4 * 3^9/2^9
= 3^5/2^6
= 243/64
To find the 10th term of the sequence, we need to look for a pattern in the terms given. Let's observe the sequence closely and try to identify any relationships or patterns.
Looking at the given terms, we can notice that to obtain the next term, we are dividing the previous term by a decreasing power of 3. Let's break it down step by step:
1. The first term, 8/81, can be obtained by dividing 8 by (3^4), which is 81. So, the numerator of the first term is decreasing by a factor of 3 at each step.
2. Similarly, the denominator of the first term is increasing by a factor of 3 at each step. Starting with the denominator 1, we multiply by 3 four times to get 81.
3. Now, let's look at the second term, 4/27. It can be obtained by dividing the numerator of the first term (8) by (3^3), which is 27. So, the numerator of the second term is obtained by dividing the previous numerator by 3.
4. Again, the denominator of the second term is obtained by multiplying the previous denominator (3) by 3.
5. We can continue this pattern for each term and keep dividing the numerator by 3 and multiplying the denominator by 3 to obtain the next term.
Now that we have identified the pattern, we can use it to find the 10th term.
To find the numerator of the 10th term, we divide the numerator of the 9th term (which is 1) by 3 ten times. 1 divided by 3 raised to the power of 10 is 1/59049.
To find the denominator of the 10th term, we multiply the denominator of the 9th term (which is 2) by 3 ten times. 2 multiplied by 3 raised to the power of 10 is 196608.
Therefore, the 10th term of the sequence is 1/59049.