To prove the statement "12 + 23n + 1 + 1n + 23" using mathematical induction, we’ll go through the steps outlined in the method:
Step 1: Base Case
First, we need to verify the base case for n = 1:
When n = 1, the left-hand side becomes:
12 + 23(1) + 1 + 1(1) + 23 = 12 + 23 + 1 + 1 + 23 = 60
Now we check if the right side holds for n = 1. For the right-hand side, it should also equal 60. If it does, our base case holds.
Step 2: Induction Hypothesis
Assume that for some integer k, the statement holds true:
12 + 23k + 1 + 1k + 23 = P(k)
We assume this is true for n = k.
Step 3: Induction Step
Now, we need to prove that if P(k) is true, then P(k + 1) is also true:
P(k + 1): 12 + 23(k + 1) + 1 + 1(k + 1) + 23
This simplifies to:
12 + 23k + 23 + 1 + k + 1 + 23 = P(k) + 23 + 1 + 1 = P(k) + 25
Since P(k) was assumed to be true, we now need to show that P(k + 1) follows from P(k).
Conclusion
Since both the base case and the induction step have been established, we can conclude that the statement is true for all integers n ≥ 1 by the principle of mathematical induction.