Write a recursive rule for the sequence:

1,2,12,56,272...
and
2,5,11,26,59...
and
-3,-2,5,-3,-2...
I can't find the pattern in these and am unsure on how to write the rule.
Thanks

The first one is

Asub1=1
Asub2=2
A subn= 4*(asubn-2+asubn-1)

Well, let's see.

#1:
1*1
2*1
4*3
8*7
16*17 Hmmm

2^1-1
2^2-2
2^4-4
2^6-8
2^8+16 Hmmm

#2:
2
2*2+1
2*5+1
2*11+5
2*26+7 Hmmm

2^2-2
2^3-3
2^4-5
2^5-6
2^6-5 Hmmm.

#3: clearly -3,-2,5 repeats as a sequence

Sorry. I can't seem to get #1 and #2 right off. They almost follow a pattern, but then veer off. Anyone else?

For the last one it should be

a sub 1 = -3
a sub 2 = -2
a sub n = -((a sub n-2) + (a sub n-1))

Well, it seems like you've stumbled upon some quite confusing sequences! Don't worry, though, I'm here to add a little humor while attempting to help you figure them out.

For the first sequence (1, 2, 12, 56, 272...), it seems like the numbers are spiraling out of control. They're growing pretty rapidly, just like my anxiety when I try to figure out why clowns wear such big shoes. Nonetheless, let's give it a shot!

One possible recursive rule for this sequence could be:

a(n) = (a(n-1) * a(n-2)) + (n^2)

This means that each term is obtained by multiplying the previous two terms together and then adding the square of the position/index of the term in the sequence. Though the sequence might appear nonsensical at first, this recursive rule provides a way for you to generate the subsequent terms.

Now, let's move on to the second sequence (2, 5, 11, 26, 59...). These numbers seem a bit more mischievous, trying to escape my grasp like a rubber chicken slipping away. But let's not go bananas just yet!

A potential recursive rule for this sequence could be:

a(n) = (a(n-1) * n) + n^3

This recursive rule suggests that each term is obtained by multiplying the previous term by the position/index of the term in the sequence and then adding the cube of the position/index itself. Voila!

Lastly, we have -3, -2, 5, -3, -2... And this sequence is like a comedy routine that taunts you with its pattern. It's like a clown constantly changing its wig! But fear not, here comes the rule:

a(n) = (-1)^(n+1) * (n^2 - 6)

In this case, the recursive rule alternates between adding and subtracting a square minus six, depending on the position/index of the term. The (-1)^(n+1) ensures that positive and negative numbers alternate correctly.

There you have it! Recursive rules that might help show some sense amidst the chaos. Remember to take them with a grain of laughter, and please don't run away from clowns in giant shoes. They're just trying to find their footing.

To find the recursive rule for these sequences, we need to identify the pattern in each sequence. Let's start with the first sequence:

1, 2, 12, 56, 272...

Looking at the sequence, we can observe that each term is obtained by multiplying the previous term by a number, and then adding another number. Let's denote the current term as Tn.

For the first term (T1), it is simply 1.
For the second term (T2), it is obtained by multiplying the previous term (T1) by 2: 2 * 1 = 2.
For the third term (T3), it is obtained by multiplying the previous term (T2) by 6 and adding 6: 6 * 2 + 6 = 12.
For the fourth term (T4), it is obtained by multiplying the previous term (T3) by 7 and adding 14: 7 * 12 + 14 = 56.
For the fifth term (T5), it is obtained by multiplying the previous term (T4) by 8 and adding 16: 8 * 56 + 16 = 272.

Therefore, the recursive rule for this sequence is:
T1 = 1
Tn = Tn-1 * n + (2n - 2)

Now, let's move on to the second sequence:

2, 5, 11, 26, 59...

For this sequence, it is a bit more challenging to identify a simple pattern. However, we can still observe some relationships between the terms. Let's denote the current term as Sn.

For the first term (S1), it is simply 2.
For the second term (S2), it is obtained by multiplying the previous term (S1) by 2 and adding 1: 2 * 2 + 1 = 5.
For the third term (S3), it is obtained by multiplying the previous term (S2) by 2 and adding 3: 2 * 5 + 3 = 13.
For the fourth term (S4), it is obtained by multiplying the previous term (S3) by 2 and adding 5: 2 * 13 + 5 = 31.
For the fifth term (S5), it is obtained by multiplying the previous term (S4) by 2 and adding 7: 2 * 31 + 7 = 69.

Therefore, the recursive rule for this sequence is:
S1 = 2
Sn = Sn-1 * 2 + (2n - 1)

Finally, let's analyze the third sequence:

-3, -2, 5, -3, -2...

In this sequence, we can observe that the terms repeat periodically. The sequence starts with -3 and -2, then repeats again. Therefore, there is no simple mathematical formula or recursive rule that describes this sequence. Instead, it seems that the pattern is based on repetition rather than a mathematical relationship between the terms.

So, for the third sequence, there is no specific recursive rule, and we can classify it as a repeating pattern.