write an explicit formula for the sequence of 1/2, 3/7, 1/3, 5/19, 3/14
rewrite it as
2/4, 3/7, 4/12, 5/19, 6/28
and it is easy to see
Tn = (n+1)/(n^2+3)
To find an explicit formula for the sequence 1/2, 3/7, 1/3, 5/19, 3/14, we need to observe the pattern within the sequence.
Let's consider the denominators of each fraction separately: 2, 7, 3, 19, 14. We notice that these denominators do not follow a simple arithmetic progression or geometric progression.
However, if we look at the numerators: 1, 3, 1, 5, 3, there seems to be a pattern. The numerators appear to be alternating in the sequence 1, 3, with the exception of the last two terms where it is 5, 3.
To create an explicit formula, we can use the pattern we observed in the numerators and a function that produces the denominators. Hence, the explicit formula for this sequence could be:
(a(n))/(b(n)) = (-1)^(n+1) * (2n-1) / (3n - 1),
where n is the term number in the sequence.
Using this formula, we can find any term in the given sequence by substituting the corresponding value of n. For example, if we substitute n = 1, we get:
(a(1))/(b(1)) = (-1)^(1+1) * (2(1)-1) / (3(1) - 1)
= 1 / 2.
Therefore, the first term of the sequence is indeed 1/2.
To find the explicit formula for this sequence, we need to look for a pattern that relates the terms. Looking at the sequence, we see that the numerator of each term alternates between the numbers 1, 3, and 5. Similarly, the denominator alternates between the numbers 2, 7, 3, 19, and 14.
Let's break down the pattern further. Notice that the numerator follows the pattern of starting with 1 and increasing by 2 with each term. We can express this pattern as (2n - 1), where n represents the position of the term in the sequence.
Next, observe that the denominator follows a different pattern. It starts with 2, then alternates between 7 and 3, and then between 19 and 14. This pattern can be expressed as 2n + k, where k follows the pattern of 5, -4, 16, -1, and so on.
Let's combine these patterns and express the explicit formula:
Numerator: 2n - 1
Denominator: 2n + k
To find the specific pattern of k, let's list the values of the sequence for n = 1, 2, 3, 4, 5:
When n = 1: 2(1) + k = 2 + k = 2k = 2
When n = 2: 2(2) + k = 4 + k = 2k = 7
When n = 3: 2(3) + k = 6 + k = 2k = 3
When n = 4: 2(4) + k = 8 + k = 2k = 19
When n = 5: 2(5) + k = 10 + k = 2k = 14
From these equations, we can determine that the pattern for k is alternating between 5 and -4. Thus, the explicit formula for the sequence is:
Term n = (2n - 1) / (2n + k)
Note: The pattern of k's could be expressed as k = (-1)^(n+1) * ((n + 5) / 2), but the formula above provides a simpler expression.