The number of non -zeroes terms in the expansion of `(sqrt(7)+1)^(75)-(sqrt(7)-1)^(75)` is

Options

A. 36
B. 37
C. 38
D. 39

Answer

To solve the question of finding the number of non-zero terms in the expansion of \((\sqrt{7}+1)^{75} - (\sqrt{7}-1)^{75}\), we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial expansion of \((a + b)^n\) is given by:
\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, we will apply this to both \((\sqrt{7}+1)^{75}\) and \((\sqrt{7}-1)^{75}\).

### Step 2: Expand Both Terms
1. **Expansion of \((\sqrt{7}+1)^{75}\)**:
\[ (\sqrt{7}+1)^{75} = \sum_{k=0}^{75} \binom{75}{k} (\sqrt{7})^{75-k} (1)^k = \sum_{k=0}^{75} \binom{75}{k} (\sqrt{7})^{75-k} \]

2. **Expansion of \((\sqrt{7}-1)^{75}\)**:
\[ (\sqrt{7}-1)^{75} = \sum_{k=0}^{75} \binom{75}{k} (\sqrt{7})^{75-k} (-1)^k \]

### Step 3: Combine the Expansions
Now, we subtract the two expansions:
\[ (\sqrt{7}+1)^{75} - (\sqrt{7}-1)^{75} = \sum_{k=0}^{75} \binom{75}{k} (\sqrt{7})^{75-k} - \sum_{k=0}^{75} \binom{75}{k} (\sqrt{7})^{75-k} (-1)^k \]

### Step 4: Simplify the Expression
This simplifies to:
\[ = \sum_{k=0}^{75} \binom{75}{k} (\sqrt{7})^{75-k} (1 - (-1)^k) \]
Notice that \(1 - (-1)^k\) is:
- \(2\) if \(k\) is odd
- \(0\) if \(k\) is even

Thus, the expression simplifies to:
\[ = 2 \sum_{k \text{ odd}} \binom{75}{k} (\sqrt{7})^{75-k} \]

### Step 5: Count the Non-Zero Terms
The non-zero terms in the expansion will only come from the odd \(k\). The odd values of \(k\) from \(0\) to \(75\) are \(1, 3, 5, \ldots, 75\).

The total number of odd integers from \(1\) to \(75\) can be calculated as follows:
- The sequence of odd numbers is \(1, 3, 5, \ldots, 75\).
- This is an arithmetic sequence where the first term \(a = 1\), the last term \(l = 75\), and the common difference \(d = 2\).

To find the number of terms \(n\) in this sequence:
\[ n = \frac{l - a}{d} + 1 = \frac{75 - 1}{2} + 1 = \frac{74}{2} + 1 = 37 + 1 = 38 \]

### Conclusion
Thus, the number of non-zero terms in the expansion of \((\sqrt{7}+1)^{75} - (\sqrt{7}-1)^{75}\) is **38**.

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