The nth term of the series
5/2,20/13,10/9,20/23.....
First of all , a series would be the sum of the terms of a sequence, you show
a sequence.
That is quite an obscure sequence, I had to resort to Wolfram to find a pattern, it showed it as
term(n) = 20/(5n+3)
http://www.wolframalpha.com/input/?i=pattern+%7B5%2F2,20%2F13,10%2F9,20%2F23,+...%7D
What methods have you learned that would let you find that ?
Answer
To find the nth term of the series 5/2, 20/13, 10/9, 20/23..., let's first observe the pattern:
- The numerator follows a pattern of 5, 20, 10, 20...
- The denominator follows a pattern of 2, 13, 9, 23...
To determine the nth term, we need to find the patterns for both the numerator and the denominator.
For the numerator:
- The pattern alternates between 5 and 10. Since n is odd for the numerator when n is even, we have 5, and when n is odd, we have 10.
For the denominator:
- The pattern is not as straightforward, as it does not follow a constant increment or decrement. However, we can see that the denominator starts with 2 and increases by 11, then decreases by 4 and increases by 14. This pattern occurs every two terms.
Now, let's address each case separately based on whether n is even or odd:
If n is even:
- The numerator is 5.
- The denominator follows the pattern 2, 13, 2 + 11, 13 + 4, 2 + 11 + 11, 13 + 4 + 14, and so on.
- To calculate this, we can observe that every even term has the same numerator and the pattern repeats every two terms. Thus, we can use the formula (n / 2) * 14 - 1 to determine the nth even term of the denominator.
If n is odd:
- The numerator is 10.
- The denominator follows the pattern 2, 9, 2 + 11, 9 - 4, 2 + 11 + 11, 9 - 4 - 14, and so on.
- To calculate this, we can observe that every odd term has the same numerator and the pattern repeats every two terms. Thus, we can use the formula ((n + 1) / 2) * 14 - 4 to determine the nth odd term of the denominator.
Therefore, the nth term of the series can be calculated as follows:
If n is even:
nth term = numerator (5) / denominator (((n / 2) * 14 - 1))
If n is odd:
nth term = numerator (10) / denominator (((n + 1) / 2) * 14 - 4)
Using these formulas, you can calculate the nth term of the series for any given value of n.