which term of the sequence 18,12,8,.... is 512/729 is ?
1. find the common ratio:
r=T(n)/T(n-1)=12/18=2/3
2. form the general sequence
(considering 18 as first term, or n=1)
T(n)=ar^(n-1)=18(2/3)^(n-1)
3. solve for
T(n)=18(2/3)^(n-1)=512/729
or even simpler,
(2/3)^(n-1)=256/6561=(2/3)^8
Solve for n.
512/729 = 2^9/3^6
= 2^8/3^8 * 2*3^2
Thanks :)
To find the term in the sequence that is equal to 512/729, we need to first figure out the pattern or rule that the sequence follows.
Looking at the given numbers, we can see that each term is obtained by dividing the preceding term by 1.5.
Let's apply this rule to find the missing term. Starting with the first term, 18, we repeatedly divide by 1.5 until we find the desired term:
18 ÷ 1.5 = 12
12 ÷ 1.5 = 8
8 ÷ 1.5 = 5.333...
5.333... ÷ 1.5 = 3.555...
3.555... ÷ 1.5 = 2.370...
2.370... ÷ 1.5 = 1.580...
1.580... ÷ 1.5 = 1.053...
We can observe that the terms are approaching a value very close to 1. Therefore, we can approximate the term to be 1.
Let's continue dividing by 1.5 one more time:
1 ÷ 1.5 = 0.666...
As the value keeps decreasing, it is clear that 512/729 lies between 0.666... and 1.
Therefore, the term in the sequence that is equal to 512/729 is the next term after 1, which is approximately 0.666..., or in fraction form, 2/3.