What is the first natural number that causes the statement:(2+3)^n<=4^n+3^n to fail?
A) 1
B) 2
C) 3
D) 4
E) 5
just start checking:
5^1 <= 4^1 + 3^1
5^2 <= 4^2 + 3^2
5^3 > 4^3 + 3^3
To find the first natural number that causes the given statement to fail, we can start by checking the options one by one.
For option A) 1:
Substituting n=1 into the statement:
(2+3)^1 <= 4^1 + 3^1
5 <= 4 + 3
5 <= 7
The statement holds true for n=1, so option A) 1 is not the correct answer.
Moving on to option B) 2:
Substituting n=2 into the statement:
(2+3)^2 <= 4^2 + 3^2
25 <= 16 + 9
25 <= 25
The statement holds true for n=2, so option B) 2 is not the correct answer.
Continuing with option C) 3:
Substituting n=3 into the statement:
(2+3)^3 <= 4^3 + 3^3
125 <= 64 + 27
125 <= 91
The statement does not hold true for n=3, so option C) 3 is a potential candidate for the correct answer.
Proceeding to option D) 4:
Substituting n=4 into the statement:
(2+3)^4 <= 4^4 + 3^4
625 <= 256 + 81
625 <= 337
The statement does not hold true for n=4, so option D) 4 is a potential candidate for the correct answer.
Lastly, let's check option E) 5:
Substituting n=5 into the statement:
(2+3)^5 <= 4^5 + 3^5
3125 <= 1024 + 243
3125 <= 1267
The statement does not hold true for n=5, so option E) 5 is a potential candidate for the correct answer.
Comparing the potential candidates, we see that option C) 3 is smaller than option D) 4 and option E) 5. Therefore, the correct answer is C) 3.