`2^(10)+2^(9) .3^1+.....2.3^9+3^(10)=A-2^(11)`. Find `A=`

Answer

To solve the equation \(2^{10} + 2^{9} \cdot 3^{1} + 2^{8} \cdot 3^{2} + \ldots + 2^{1} \cdot 3^{9} + 3^{10} = A - 2^{11}\), we can follow these steps:

### Step 1: Identify the Series
The left-hand side of the equation can be expressed as a sum of terms where each term is of the form \(2^{10-k} \cdot 3^{k}\) for \(k = 0\) to \(10\). Thus, we can rewrite the series as:
\[ S = \sum_{k=0}^{10} 2^{10-k} \cdot 3^{k} \]

### Step 2: Factor Out \(2^{10}\)
We can factor out \(2^{10}\) from the series:
\[ S = 2^{10} \sum_{k=0}^{10} \left(\frac{3}{2}\right)^{k} \]

### Step 3: Recognize the Geometric Series
The sum \(\sum_{k=0}^{10} \left(\frac{3}{2}\right)^{k}\) is a geometric series with the first term \(a = 1\) and common ratio \(r = \frac{3}{2}\). The number of terms \(n = 11\).

### Step 4: Use the Formula for the Sum of a Geometric Series
The formula for the sum of the first \(n\) terms of a geometric series is:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Applying this formula:
\[ \sum_{k=0}^{10} \left(\frac{3}{2}\right)^{k} = \frac{1 - \left(\frac{3}{2}\right)^{11}}{1 - \frac{3}{2}} = \frac{1 - \left(\frac{3}{2}\right)^{11}}{-\frac{1}{2}} = -2 \left(1 - \left(\frac{3}{2}\right)^{11}\right) \]

### Step 5: Substitute Back into the Expression for \(S\)
Substituting this back into our expression for \(S\):
\[ S = 2^{10} \cdot -2 \left(1 - \left(\frac{3}{2}\right)^{11}\right) = -2^{11} \left(1 - \left(\frac{3}{2}\right)^{11}\right) \]

### Step 6: Simplify the Expression
Now, we can simplify:
\[ S = -2^{11} + 2^{11} \cdot \left(\frac{3}{2}\right)^{11} = -2^{11} + \frac{3^{11}}{2^{11}} \cdot 2^{11} = -2^{11} + 3^{11} \]

### Step 7: Set Equal to \(A - 2^{11}\)
From the original equation:
\[ -2^{11} + 3^{11} = A - 2^{11} \]
Adding \(2^{11}\) to both sides gives:
\[ 3^{11} = A \]

### Final Answer
Thus, we find that:
\[ A = 3^{11} \]