Two consecutive terms in the expansion of `(3+2x)^74` have equal coefficients then term are (A) `30 and 31` (B) 38 and 39 (C) 31 and 32 (D) 37 and 38

Options

A. `7^(th) and 8^(th)`
B. `11^(th) and 12(th)`
C. `30^(th) and 31^(th)`
D. `31^(th) and 32^(th)`

Answer

To solve the problem of finding two consecutive terms in the expansion of \((3 + 2x)^{74}\) that have equal coefficients, we can follow these steps:

### Step 1: Identify the General Term
The general term \(T_r\) in the expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For our case, \(a = 3\), \(b = 2x\), and \(n = 74\). Therefore, the \(r\)-th term is:
\[ T_{r+1} = \binom{74}{r} (3)^{74-r} (2x)^r \]

### Step 2: Write the Consecutive Terms
The two consecutive terms are:
- \(T_{r+1} = \binom{74}{r} (3)^{74-r} (2x)^r\)
- \(T_{r+2} = \binom{74}{r+1} (3)^{74-(r+1)} (2x)^{r+1}\)

### Step 3: Set Up the Equation for Equal Coefficients
We need to find \(r\) such that the coefficients of these two terms are equal:
\[ \binom{74}{r} (3)^{74-r} (2^r) = \binom{74}{r+1} (3)^{73-r} (2^{r+1}) \]

### Step 4: Simplify the Equation
Using the property of binomial coefficients, \(\binom{n}{r+1} = \frac{n-r}{r+1} \binom{n}{r}\), we can rewrite the equation:
\[ \binom{74}{r} (3)^{74-r} (2^r) = \frac{74-r}{r+1} \binom{74}{r} (3)^{73-r} (2^{r+1}) \]
Cancelling \(\binom{74}{r}\) from both sides (assuming \(r \neq 74\)), we get:
\[ (3)^{74-r} (2^r) = \frac{74-r}{r+1} (3)^{73-r} (2^{r+1}) \]

### Step 5: Further Simplification
Dividing both sides by \((3)^{73-r}\) and \(2^r\):
\[ 3 = \frac{74-r}{r+1} \cdot 2 \]
This simplifies to:
\[ 3(r+1) = 2(74-r) \]

### Step 6: Solve for \(r\)
Expanding and rearranging gives:
\[ 3r + 3 = 148 - 2r \]
\[ 5r = 145 \implies r = 29 \]

### Step 7: Find the Terms
Now, we need to find the two consecutive terms:
- \(T_{30} = T_{r+1} = T_{29+1}\)
- \(T_{31} = T_{r+2} = T_{29+2}\)

Thus, the terms are \(T_{30}\) and \(T_{31}\).

### Conclusion
The two consecutive terms in the expansion of \((3 + 2x)^{74}\) that have equal coefficients are:
\[ \text{Terms: } 30 \text{ and } 31 \]
Thus, the answer is (A) 30 and 31.

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