write a rule for nth term:
1/10, 3/20, 5/30, 7/40,....
i think the next term is 9/50 but i don't know how i got that
[2n-n^0]/10n
Well, don't worry! I'm here to help you figure it out with a touch of humor.
Let's take a closer look at the given sequence: 1/10, 3/20, 5/30, 7/40, ...
First and foremost, let's focus on the denominators. Notice that they're increasing by 10 with each term. So, it's safe to say that the nth denominator is given by 10n. Why? Well, maybe the sequence just had a thing for even multiples of 10. Who doesn't, right?
Now, let's move on to the numerators. They're progressing in an odd manner: 1, 3, 5, 7. It seems like someone just forgot about all the even numbers, but hey, we don't judge here!
In simpler terms, you can see that the numerators are simply 2n - 1. It's like they're protesting against evenness and advocating for the odd numbers' rights!
Now, combining these two ideas, we can say that the nth term of this sequence is (2n - 1)/(10n).
So, when n = 5, the 5th term would be (2 * 5 - 1) / (10 * 5), which simplifies to 9/50. Great job, your gut instinct was right! Just remember, when it comes to math, it's always good to follow your nose.
To find the rule for the nth term of the given sequence: 1/10, 3/20, 5/30, 7/40, ...
Step 1: Observe the pattern in the numerators: 1, 3, 5, 7, ...
We can see that the numerators are increasing by 2 each time.
Step 2: Observe the pattern in the denominators: 10, 20, 30, 40, ...
We can see that the denominators are increasing by 10 each time.
Step 3: Write the nth term rule for the numerator:
n * 2
The formula "n * 2" gives us the nth term for the numerators.
Step 4: Write the nth term rule for the denominator:
n * 10
The formula "n * 10" gives us the nth term for the denominators.
Step 5: Combine the nth term rules for the numerator and denominator:
For the numerator: n * 2
For the denominator: n * 10
So, the nth term of the given sequence is:
(n * 2) / (n * 10)
Simplifying this expression, we get:
1/10, 3/20, 5/30, 7/40, ...
This confirms that the next term in the sequence would be 9/50.
To find the rule for the nth term of the given sequence, let's examine the pattern and break it down step by step:
1/10, 3/20, 5/30, 7/40, ...
First, let's simplify each fraction by dividing the numerator and denominator by their greatest common divisor (GCD) to get rid of any common factors:
1/10 = 1/10
3/20 = 3/10
5/30 = 1/6
7/40 = 7/40
...
Now, let's observe the pattern between the numerators and denominators:
Numerator: 1, 3, 1, 7, ...
Denominator: 10, 10, 6, 40, ...
From the numerator sequence, we can see that it alternates between 1 and 3.
From the denominator sequence, we can notice that it increases by 0 for the first two terms, then by 4 for the third term, and finally by 30 for the fourth term.
Putting these observations together, we can establish the rule for the nth term as follows:
Numerator: 2n - 1
Denominator: 10 + (n-2)*(n-1)*15
Using this rule, let's find the value for the next term, which should be the 5th term:
Numerator: 2*5 - 1 = 9
Denominator: 10 + (5-2)*(5-1)*15 = 50
So, the next term in the sequence is indeed 9/50.