Randy had a sheep that started eating more and more as each day passed. on the first day, Tandy's sheep ate 5 carrots. On the second day, Randy's sheep ate 11 carrots. On the third day, the sheep ate 18 carrots. And on the fourth day, the gluttonous sheep ate 26 carrots. How many carrots will the sheep eat on the 197th day?

Question

Randy had a sheep that started eating more and more as each day passed. on the first day, Tandy's sheep ate 5 carrots. On the second day, Randy's sheep ate 11 carrots. On the third day, the sheep ate 18 carrots. And on the fourth day, the gluttonous sheep ate 26 carrots. How many carrots will the sheep eat on the 197th day?

5+6+11+7+18+8=26.
an^2+bn+c=-2
n=1, a+b+c=2
n=2, 4a+2b+c=4
n=3, 9a+6b+c=7

Answer 1

so we have this pattern:

1 5
2 11
3 18
4 26

notice that on each consecutive day, the increase is one more than the previous day's increase, suggesting a quadratic expression

let number = an^2 + bn + c , were n is the number of days
if n = 1: a+b+c = 5
if n = 2: 4a + 2b + c = 11
if n = 3: 9a + 3b + c = 18

2nd - first: 3a + b = 6
third - 2nd : 5a + b = 7
subtract those last two:
2a = 1
a = 1/2

plug this into 3a+b=6
3/2 + b = 6
b = 6-3/2 = 9/2
back into 1st
a+b+c=5
1/2 + 9/2 + c = 5
c = 0

so number = (1/2)n2+ (9/2)n
= (n^2 + 9n)/2

so if n = 197
number = (197^2 + 9(197))/2 = 20291

Answer 2

To find the pattern in the number of carrots eaten by Randy's sheep each day, it seems like there is a pattern of adding consecutive odd numbers.

On the first day, the sheep ate 5 carrots.
On the second day, the sheep ate 5 + 6 = 11 carrots.
On the third day, the sheep ate 5 + 6 + 7 = 18 carrots.
On the fourth day, the sheep ate 5 + 6 + 7 + 8 = 26 carrots.

By observing the pattern, we can see that each day the sheep is eating consecutive odd numbers starting from 5.

To find out how many carrots the sheep will eat on the 197th day, we need to find the 197th consecutive odd number after 5 and then sum all the numbers from 5 to that consecutive odd number.

To find the 197th consecutive odd number, we can use the formula:

nth odd number = 2n - 1, where n is the position of the number in the sequence.

So, the 197th consecutive odd number would be:
2 * 197 - 1 = 393 - 1 = 392

Now, to find the sum of the numbers from 5 to 392, we can use the formula for the sum of consecutive integers:

Sum = (n/2)(first term + last term), where n is the number of terms.

In this case:
n = 392 - 5 + 1 = 388
first term = 5
last term = 392

Plugging these values into the formula, we get:
Sum = (388/2)(5 + 392) = 194(397) = 76918

Therefore, the sheep will eat 76918 carrots on the 197th day.