find the n th term of the series 5/2+20/13+10/9+20/23..... .
Just like in your previous problem, the corresponding AS is
2/5 + 13/20 + 9/10 + 23/20 + ...
= 8/20 + 13/20 + 18/20 + 23/20 + ...
looks like a = 8/20, and d = 5/20
or a = 2/5 and d = 1/4
so term(n) of this AS
= a + (n-1)d
= 2/5 +(1/4)(n-1)
= 2/5 + n/4 - 1/4
= n/4 + 3/20
= (5n + 3)/20
and finally the general term of the given HS
= 20/(5n+3)
20/5n+3
Ah, series and terms, so fancy! Let me put on my math clown hat and calculate this for you.
Now, if I understood correctly, you need the "n-th" term of the series. Well, this sounds like a job for clownonomics!
Let's first take a closer look at the series: 5/2, 20/13, 10/9, 20/23... Is there some kind of secret code going on here? Maybe it's the clown code!
Upon closer inspection, I noticed that the numerator and denominator seem to alternate. It's like they are doing a little dance! So, if we codify this dance, we can see a pattern emerging.
The numerator goes like 5, 20, 10, 20... That's easy! It keeps alternating between 5 and 20.
Now, for the denominator, we have 2, 13, 9, 23... Hmm, this one seems a bit trickier. But fear not, my friend! There is a simple formula to this madness. It seems that the denominator is progressing by adding the previous number to twice its value.
So, we can say that the n-th term of the series is:
Numerator: If n is odd, it's 5. If n is even, it's 20.
Denominator: If n is 1, it's 2. For all other n, it's (previous denominator) + 2*(previous denominator).
Now, you can just plug in the value of n into these formulas, and you'll have your answer. Who said math couldn't be fun? Enjoy crunching those numbers, my mathematical clown comrade!
To find the nth term of the series 5/2 + 20/13 + 10/9 + 20/23 + ..., we need to first observe the pattern in the series.
Upon closer inspection, we can see that the numerators alternate between 5 and 20, while the denominators follow the pattern 2, 13, 9, 23, ... . It seems that the denominators are increasing by 11 after every two terms.
Using this information, we can determine the general formula for the nth term by creating separate formulas for the numerators and denominators.
1. Numerator formula:
- When n is odd: 5
- When n is even: 20
2. Denominator formula:
- When n is odd: 2 + (n // 2) * 11
- When n is even: 13 + (n // 2 - 1) * 11
Note: "//" denotes integer division.
Now we can combine these formulas to determine the nth term:
- When n is odd: (5) / (2 + (n // 2) * 11)
- When n is even: (20) / (13 + (n // 2 - 1) * 11)
Make sure to check whether the numerator or denominator formulas apply based on the parity of n.
To find the nth term of the series 5/2+20/13+10/9+20/23..., we can observe a pattern and try to identify a formula.
Looking at the terms of the series, we can see that the numerators alternate between 5 and 20, and the denominators increase by 4 each time: 2, 13, 9, 23, ...
To obtain a formula for the numerators, notice that the pattern alternates between 5 and 20. We can represent this using the formula (-1)^n × 15, where n represents the position of the term in the series (0, 1, 2, 3, ...).
For the denominators, we can notice that each term is obtained by adding 4 to the previous term, starting from 2. We can represent this using the formula 9 + 4n, where n represents the position of the term in the series (0, 1, 2, 3, ...).
Now, we can combine these formulas to find the nth term of the series:
Numerator formula: (-1)^n × 15
Denominator formula: 9 + 4n
Using these formulas, the nth term of the series can be represented as:
(-1)^n × 15 / (9 + 4n)
So, the general formula for the nth term of the given series is (-1)^n × 15 / (9 + 4n).