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

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