Need help with this one
write an explicit formula for the sequence 1/2,3//7,1/3,5/19,3/14 then find q14
I don't know how to set this up can someone please show me how this one works
To find the explicit formula for the given sequence, we need to identify the pattern or rule that governs the sequence.
Let's look at the terms of the sequence:
1st term: 1/2
2nd term: 3/7
3rd term: 1/3
4th term: 5/19
5th term: 3/14
To identify the pattern, let's focus on the numerators and denominators separately:
Numerators: 1, 3, 1, 5, 3
Denominators: 2, 7, 3, 19, 14
Looking at the numerators, notice that we have a pattern where the odd-indexed terms (1st, 3rd, 5th) are increasing by 2 each time:
1, _, 3, _, 5
To represent this pattern mathematically, we can use the formula (2n - 1), where n represents the term number.
Looking at the denominators, notice that we have a pattern where the terms are increasing by different numbers. The increase values seem to be alternating between 5 and 9. So, we can represent the denominator pattern as 7n - 4 when n is odd and 5n + 9 when n is even.
Combining these patterns, we can write the explicit formula for the sequence as:
an = (2n - 1) / (7n - 4) for odd-indexed terms (1st, 3rd, 5th, ...)
an = (2n - 1) / (5n + 9) for even-indexed terms (2nd, 4th, 6th, ...)
Now, let's find the 14th term (q14) using the explicit formula:
Since 14 is an even number, we will use the formula for the even-indexed terms:
a14 = (2 * 14 - 1) / (5 * 14 + 9)
Calculating this expression:
a14 = (28 - 1) / (70 + 9)
a14 = 27 / 79
Therefore, the 14th term of the sequence is 27/79.