Consider the expansion of `(2x-1)^(15)`

Options

A. Algebraically least coefficient is `T_(5)`
B. Algebraically greatest coefficient is `T_(5)`
C. Numerically greatest coefficient is `T_(6)`
D. Numerically least coefficient is `T_(16)`

Explanation

To solve the problem of finding the algebraically least coefficient in the expansion of \((2x - 1)^{15}\), we will follow these steps:

### Step 1: Identify the general term in the binomial expansion
The general term \(T_r\) in the expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 2x\), \(b = -1\), and \(n = 15\). Therefore, the general term becomes:
\[ T_r = \binom{15}{r} (2x)^{15-r} (-1)^r \]

### Step 2: Simplify the general term
Substituting the values into the general term:
\[ T_r = \binom{15}{r} (2^{15-r} x^{15-r}) (-1)^r \]
This simplifies to:
\[ T_r = \binom{15}{r} (-1)^r 2^{15-r} x^{15-r} \]

### Step 3: Determine the algebraically least coefficient
To find the algebraically least coefficient, we need to analyze the coefficients of the terms. The coefficient of \(x^{15-r}\) is:
\[ \text{Coefficient} = \binom{15}{r} (-1)^r 2^{15-r} \]
The sign of the coefficient depends on \((-1)^r\). The coefficients will alternate in sign based on the value of \(r\).

### Step 4: Find the value of \(r\) that gives the least coefficient
The absolute value of the coefficient is:
\[ |\text{Coefficient}| = \binom{15}{r} 2^{15-r} \]
To find the least algebraic coefficient, we need to find the maximum value of \(|\text{Coefficient}|\) when \(r\) is odd (since the least coefficient will be negative).

### Step 5: Calculate the coefficients for odd \(r\)
We will calculate \(|\text{Coefficient}|\) for odd values of \(r\) (1, 3, 5, 7, 9, 11, 13, 15):
- For \(r = 1\):
\[ |\text{Coefficient}| = \binom{15}{1} 2^{14} = 15 \cdot 16384 = 245760 \]
- For \(r = 3\):
\[ |\text{Coefficient}| = \binom{15}{3} 2^{12} = 455 \cdot 4096 = 1863680 \]
- For \(r = 5\):
\[ |\text{Coefficient}| = \binom{15}{5} 2^{10} = 3003 \cdot 1024 = 3075072 \]
- For \(r = 7\):
\[ |\text{Coefficient}| = \binom{15}{7} 2^{8} = 6435 \cdot 256 = 1658880 \]
- For \(r = 9\):
\[ |\text{Coefficient}| = \binom{15}{9} 2^{6} = 5005 \cdot 64 = 320320 \]
- For \(r = 11\):
\[ |\text{Coefficient}| = \binom{15}{11} 2^{4} = 1365 \cdot 16 = 21840 \]
- For \(r = 13\):
\[ |\text{Coefficient}| = \binom{15}{13} 2^{2} = 105 \cdot 4 = 420 \]
- For \(r = 15\):
\[ |\text{Coefficient}| = \binom{15}{15} 2^{0} = 1 \]

### Step 6: Identify the least coefficient
From the calculations, the least coefficient occurs when \(r = 5\) (which gives the maximum absolute value) and is negative:
\[ \text{Least coefficient} = -3075072 \]

### Conclusion
The algebraically least coefficient in the expansion of \((2x - 1)^{15}\) is \(-3075072\).