Final Answer:
To prove that (2^n = 4) using mathematical induction, we need to show that the statement is true for all positive integers (n). Thus, we have shown that (2^n = 4) using mathematical induction.
Step-by-step explanation:
Basis Step:
Let’s start by verifying the statement for the smallest possible value of (n), which is (n=1).
Substituting (n=1) in the statement, we get:
2^1=4
which is true. Therefore, the statement is true for (n=1).
Induction Step:
Now, we need to show that if the statement is true for some positive integer (k), then it must also be true for (k+1).
Assuming that the statement is true for (k), we have:
2^k=4
Multiplying both sides by 2, we get:
2^(k+1)=2⋅4=8
Therefore, the statement is true for (k+1).
Since the statement is true for (n=1) and if it is true for some positive integer (k), then it must also be true for (k+1), we can conclude that the statement is true for all positive integers (n).
Thus, we have shown that (2^n = 4) using mathematical induction.