what is the nth term in the sequence 12,6,4,3,2,12/5
One can quickly see that it is neither arithmetic nor geometric in nature,
nor is it harmonic.
I took several levels of "differences" and found no common difference, so
no simple polynomial expresses it
Not even Wolfram could find a pattern:
https://www.wolframalpha.com/input/?i=pattern+%7B12,6,4,3,2,12%2F5%7D
I tried a "best fit" and got this:
https://www.wolframalpha.com/input/?i=best+fit+%7B12,6,4,3,2,12%2F5%7D
I give up
Consider the ratio of successive terms. These ratios are
1/2, 2/3, 3/4, 4/6, 6/5
It was looking pretty good till the last two...
In this case, may be it best to use interpolation.
In wolframalpha. c o m paste:
interpolate ( 1 , 12 ) , ( 2 , 6 ) , ( 3 , 4 ) , ( 4 , 3 ) , ( 5 , 2 ) , ( 6, 12 / 5 )
interpolating polynomial is:
x ^ 5 / 300 + x ^ 4 / 30 - 21 x ^ 3 / 20 + 43 x ^ 2 / 6 - 3113 x / 150 + 133 / 5
OR
1 / 300 ( x ^ 5 + 10 x ^ 4 - 315 x ^ 3 + 2150 x ^ 2 - 6226 x + 7980 )
Replace x with n.
an = 1 / 300 ( n ^ 5 + 10 n ^ 4 - 315 n ^ 3 + 2150 n ^ 2 - 6226 n + 7980 )
To find the nth term in a sequence, we need to identify the pattern or rule that governs the sequence.
Looking at the sequence: 12, 6, 4, 3, 2, 12/5
We can observe that each term is obtained by dividing the previous term by a decreasing value of 2.
To be precise, the first term, 12, is divided by 2 to get the second term, 6. Then, 6 is divided by 2 to get the third term, 4. Continuing this pattern, each term is divided by 2 to obtain the next term.
However, the last term, 12/5, seems to break this pattern.
To determine the rule for this sequence, we can rewrite the terms using exponents:
12 = 2^3
6 = 2^2
4 = 2^2
3 = 2^1
2 = 2^1
12/5 = 2^-1
We can see that all the terms can be written as 2 raised to a power, except for the last term, which is a negative exponent.
Therefore, the rule for the nth term in this sequence is:
2^(3-(n-1)) = 2^(4-n), for n ∈ ℕ
For example, if we want to find the 5th term, we substitute n = 5 into the rule:
2^(4-5) = 2^(-1) = 1/2
Hence, the 5th term in the sequence is 1/2.