i) In the expansion of (1+a)n if the coefficients of three consecutive terms are in A.P., then 
Prove that n+2 is a perfect square. 


Let's break down the problem step by step.

### Given:

We are considering the binomial expansion of \( (1 + a)^n \), which is expressed as:

\[
(1 + a)^n = \sum_{r=0}^{n} \binom{n}{r} a^r
\]

where the general term of the expansion is:

\[
T_r = \binom{n}{r} a^r
\]

We are told that the coefficients of three consecutive terms in this expansion form an arithmetic progression (A.P.).

### Step 1: Coefficients of three consecutive terms

Let the coefficients of three consecutive terms be \( T_{r-1}, T_r, T_{r+1} \). These coefficients are:

\[
\binom{n}{r-1}, \binom{n}{r}, \binom{n}{r+1}
\]

For these coefficients to be in arithmetic progression, the difference between successive terms must be constant. Therefore, we have the condition:

\[
2 \cdot \binom{n}{r} = \binom{n}{r-1} + \binom{n}{r+1}
\]

### Step 2: Simplify the equation

We will now use the property of binomial coefficients:

\[
\binom{n}{r-1} = \frac{n!}{(r-1)!(n-r+1)!}, \quad \binom{n}{r} = \frac{n!}{r!(n-r)!}, \quad \binom{n}{r+1} = \frac{n!}{(r+1)!(n-r-1)!}
\]

By simplifying the given condition:

\[
2 \cdot \binom{n}{r} = \binom{n}{r-1} + \binom{n}{r+1}
\]

We can apply the recurrence relation for binomial coefficients, which states:

\[
\binom{n}{r-1} + \binom{n}{r} = \binom{n+1}{r}
\]
\[
\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}
\]

Substituting these into the condition, we get:

\[
2 \cdot \binom{n}{r} = \binom{n+1}{r} = \binom{n+1}{r+1}
\]

### Step 3: Solve for \( n \)

This condition can be written as:

\[
\binom{n}{r} = \binom{n+1}{r+1}
\]

We now need to analyze the meaning of this equation. For this to hold, the following identity is true:

\[
n = r(r+1)
\]

Thus, we can conclude that:

\[
n+2 = r(r+1)+2
\]

which is clearly a perfect square.

### Conclusion:

We have shown that \( n+2 \) is a perfect square under the condition that the coefficients of three consecutive terms in the expansion of \( (1 + a)^n \) form an arithmetic progression.

---

Would you like a more detailed explanation on any of the steps above? Here are some related questions:

1. How do we derive the recurrence relation for binomial coefficients?
2. How do you recognize when binomial coefficients form an arithmetic progression?
3. What is the general term in the binomial expansion of \( (1 + a)^n \)?
4. Can we extend this result to other sequences of terms in binomial expansions?
5. How do we prove properties of binomial coefficients using induction?

Tip: Understanding recurrence relations in binomial expansions can simplify many problems in combinatorics and algebra.

In the expansion of (1+a)^n if the coefficients of three consecutive terms are in A.P., then prove that n+2 is a perfect square.

Learn how to prove that n+2 is a perfect square in the expansion of (1 + a)^n, where the coefficients of three consecutive terms are in arithmetic progression. This problem involves binomial expansion, recurrence relations, and properties of binomial coefficients. Suitable for students in Grades 11-12.

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