you have a pair of newborn hamsters. each pair that is at least one month old produces a new pair every month. at this rate, you will have 178 hamsters at the end of what month? (The math relay officials said the 10th month but I think that it should be the 8th)
so
2 + 4 + 8 + 16 + ... = 178
sum(n) = a(r^n - 1)/(r-1)
2(2^n - 1)/1 = 178
2^n - 1 = 89
2^n = 90
I know 2^6 = 64
2^7 = 128
2^8 = 256
So I agree with your answer ,
at the end of 7 you only have 128 but you need 178
so it must be the 8th month
how did they come up with 10?
perhaps they interpreted their own question incorrectly and found the term number close to 178
let's see which term number is close to 178
term(n) = ar^(n-1)
2(2^(n-1)) = 178
2^(n-1) = 98
n-1 = ln 98/ln2
= 6.6 not anywhere near 10
so , they must be wrong.
To solve this problem, we need to understand the breeding pattern of the hamsters.
Starting with a pair of newborn hamsters, each pair becomes capable of reproduction after one month. This means that in the second month, the original pair can produce a new pair. In the third month, the original pair can still produce another pair, and the new pair from the second month can also produce a pair. This pattern continues, with each pair from the previous month being capable of reproduction.
Let's break down this breeding pattern month by month:
Month 1: 1 pair
Month 2: 1 pair (from the previous month) + 1 new pair = 2 pairs
Month 3: 2 pairs (from the previous month) + 1 new pair + 1 new pair = 4 pairs
Month 4: 4 pairs (from the previous month) + 2 new pairs + 1 new pair = 7 pairs
Month 5: 7 pairs (from the previous month) + 3 new pairs + 2 new pairs = 12 pairs
Month 6: 12 pairs (from the previous month) + 5 new pairs + 3 new pairs = 20 pairs
Month 7: 20 pairs (from the previous month) + 8 new pairs + 5 new pairs = 33 pairs
Month 8: 33 pairs (from the previous month) + 13 new pairs + 8 new pairs = 54 pairs
Month 9: 54 pairs (from the previous month) + 21 new pairs + 13 new pairs = 88 pairs
Month 10: 88 pairs (from the previous month) + 34 new pairs + 21 new pairs = 143 pairs
Month 11: 143 pairs (from the previous month) + 55 new pairs + 34 new pairs = 232 pairs
Month 12: 232 pairs (from the previous month) + 89 new pairs + 55 new pairs = 376 pairs
Based on this pattern, we can see that at the end of the 8th month, we will have 54 pairs of hamsters, which is a total of 108 hamsters.
Therefore, the answer should be the 9th month, not the 8th month.