No. of terms in the expansion of `(1+2x)^(9) +(1-2x)^(9)` is :

Options

A. 10
B. 9
C. 7
D. 5

Explanation

To find the number of terms in the expansion of \( (1 + 2x)^9 + (1 - 2x)^9 \), we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial expansion of \( (a + b)^n \) is given by:
\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our case, we have two expansions: \( (1 + 2x)^9 \) and \( (1 - 2x)^9 \).

### Step 2: Expand Both Terms
1. The expansion of \( (1 + 2x)^9 \) will yield terms of the form:
\[ \binom{9}{k} (1)^{9-k} (2x)^k = \binom{9}{k} 2^k x^k \]
for \( k = 0, 1, 2, \ldots, 9 \).

2. The expansion of \( (1 - 2x)^9 \) will yield terms of the form:
\[ \binom{9}{k} (1)^{9-k} (-2x)^k = \binom{9}{k} (-2)^k x^k \]
for \( k = 0, 1, 2, \ldots, 9 \).

### Step 3: Combine the Two Expansions
When we add the two expansions together:
\[ (1 + 2x)^9 + (1 - 2x)^9 \]
the terms with odd powers of \( x \) will cancel out because they will have opposite signs. The terms with even powers will add up.

### Step 4: Identify the Even Powers
The even powers of \( x \) in the expansion are:
- \( x^0 \) (constant term)
- \( x^2 \)
- \( x^4 \)
- \( x^6 \)
- \( x^8 \)

### Step 5: Count the Number of Terms
The even powers of \( x \) from \( 0 \) to \( 8 \) are \( 0, 2, 4, 6, 8 \). This gives us a total of:
- 5 terms.

### Final Answer
Thus, the number of terms in the expansion of \( (1 + 2x)^9 + (1 - 2x)^9 \) is **5**.

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