50. What is wrong with this "proof"?
"Theorem" For every positive integer n, [i=, i = (n + 2) /2.
Basis Step: The formula is true for n = 1.
Inductive Step: Suppose that [i=, i = (n+ 2) /2.
Then Lit| i = (Li_i) + (n + 1). By the inductive
hypothesis,
i=1
i=(+½)212+nt1=
(n2 + n+ 4) /2 + n+ 1 = (n? + 3n + %)/2 =
(n + 2) /2 = [(n+ 1) + 21/2, completing the induc-
tive step.
51. What is wrong with

Let's break down the reasoning of the "proof" and analyze what is wrong with it. The proof seems to attempt to use mathematical induction to prove the formula:

\[
\sum_{i=1}^{n} i = \frac{n + 2}{2}
\]

This formula is already incorrect because the sum of the first \(n\) positive integers is actually given by the well-known formula:

\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]

### 1. **The Incorrect Formula**
The formula provided in the "proof" states that:

\[
\sum_{i=1}^{n} i = \frac{n + 2}{2}
\]

This is clearly incorrect because it simplifies the sum of an arithmetic series in an invalid way. For example:

- When \(n = 1\), the sum \(\sum_{i=1}^{1} i = 1\), but according to the formula in the "proof," we get:
  \[
  \frac{1 + 2}{2} = \frac{3}{2} \quad (\text{which is incorrect since the sum is 1, not } \frac{3}{2}).
  \]
  
Thus, the formula is wrong from the very beginning.

### 2. **Inductive Hypothesis**
The inductive hypothesis is written as:

\[
\sum_{i=1}^{n} i = \frac{n + 2}{2}
\]

This assumption is incorrect, and the proof tries to proceed based on this incorrect assumption.

### 3. **Inductive Step Misinterpretation**
The proof then tries to show that if the formula holds for \(n\), it also holds for \(n+1\). It states something like:

\[
\sum_{i=1}^{n+1} i = \sum_{i=1}^{n} i + (n + 1)
\]

This part is correct, but then the proof uses the incorrect inductive hypothesis:

\[
\sum_{i=1}^{n} i = \frac{n + 2}{2}
\]

to substitute for the sum up to \(n\). This leads to incorrect results and algebraic manipulation.

### 4. **Conclusion**
The major issue is that the original formula is wrong. The correct formula for the sum of the first \(n\) integers is \(\frac{n(n + 1)}{2}\), not \(\frac{n + 2}{2}\). This invalidates the entire proof, including both the base case and the inductive step.

### To summarize, the main mistakes are:
1. The formula being proved is incorrect.
2. The inductive hypothesis assumes the incorrect formula.
3. The algebra in the inductive step is based on this incorrect assumption.

Would you like more details on mathematical induction or summation formulas? 

Here are 5 related questions to deepen understanding:

1. What is the correct formula for the sum of the first \(n\) integers, and how is it derived?
2. How does mathematical induction work, and when is it used in proving statements?
3. Can you prove the correct summation formula \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\) using induction?
4. What are common pitfalls to avoid when using mathematical induction?
5. Can you explain the difference between arithmetic and geometric series and how their sums are derived?

**Tip:** When using mathematical induction, always ensure that the base case and the inductive step are both correct and consistent with the formula you're trying to prove.

What is wrong with this "proof"? 'Theorem' For every positive integer n, [i=, i = (n + 2) /2.

Explore the errors in a flawed proof that incorrectly attempts to prove a summation formula using mathematical induction. Understand the correct formula for the sum of the first n integers and why the inductive step fails. Suitable for high school students in Grades 9-12.

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