Write down and simplify
14th term in `(3 +x)^15`

Answer

To find the 14th term in the expansion of \((3 + x)^{15}\), we can use the Binomial Theorem. The general term (or \(r+1\)th term) in the expansion of \((a + b)^n\) is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

Where:
- \(n\) is the power of the binomial,
- \(a\) and \(b\) are the terms in the binomial,
- \(r\) is the term number minus one (since we start counting from 0).

### Step 1: Identify the values
In our case:
- \(n = 15\)
- \(a = 3\)
- \(b = x\)

To find the 14th term, we need to set \(r = 13\) (since \(r + 1 = 14\)).

### Step 2: Apply the formula
Now we can substitute these values into the formula:

\[ T_{14} = \binom{15}{13} (3)^{15-13} (x)^{13} \]

### Step 3: Simplify the expression
Calculating the binomial coefficient:

\[ \binom{15}{13} = \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 \]

Now substituting back into the term:

\[ T_{14} = 105 \cdot (3)^2 \cdot (x)^{13} \]

Calculating \(3^2\):

\[ 3^2 = 9 \]

Thus, we have:

\[ T_{14} = 105 \cdot 9 \cdot x^{13} \]

### Step 4: Final multiplication
Now, multiply \(105\) and \(9\):

\[ 105 \cdot 9 = 945 \]

So, the 14th term is:

\[ T_{14} = 945x^{13} \]

### Final Answer
The 14th term in the expansion of \((3 + x)^{15}\) is:

\[ \boxed{945x^{13}} \]