If 1.3+2.`3^(2)+3.3^(3)+ . . .=n3^(n)=((2n-1)3+b)/(4)`, then a and b are respectively:

Options

A. n,2
B. n,3
C. n+1,2
D. n+1,3

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) in the equation:

\[ 1 \cdot 3^1 + 2 \cdot 3^2 + 3 \cdot 3^3 + \ldots + n \cdot 3^n = \frac{(2n - 1)3^n + b}{4} \]

### Step-by-Step Solution:

1. **Identify the Series**:
The left-hand side of the equation is a series of the form:
\[ S_n = 1 \cdot 3^1 + 2 \cdot 3^2 + 3 \cdot 3^3 + \ldots + n \cdot 3^n \]
This series is a combination of an arithmetic progression (AP) and a geometric progression (GP), known as an arithmetic-geometric progression (AGP).

2. **Use the Formula for AGP**:
The sum of the first \( n \) terms of an AGP can be expressed as:
\[ S_n = \frac{a - (a + (n - 1)d) r^n}{1 - r} + \frac{d r (1 - r^{n-1})}{(1 - r)^2} \]
where \( a \) is the first term, \( d \) is the common difference, and \( r \) is the common ratio.

3. **Identify Parameters**:
- The first term \( a = 0 \cdot 3^0 = 0 \) (since the first term is 0).
- The common difference \( d = 1 \) (the difference between successive terms).
- The common ratio \( r = 3 \).

4. **Substituting Values**:
Substituting \( a = 0 \), \( d = 1 \), and \( r = 3 \) into the AGP formula:
\[ S_n = \frac{0 - (0 + (n - 1) \cdot 1) \cdot 3^n}{1 - 3} + \frac{1 \cdot 3 (1 - 3^{n-1})}{(1 - 3)^2} \]

5. **Simplifying the Expression**:
- The first term simplifies to:
\[ S_n = \frac{-(n - 1) \cdot 3^n}{-2} = \frac{(n - 1) \cdot 3^n}{2} \]
- The second term simplifies to:
\[ \frac{3(1 - 3^{n-1})}{4} = \frac{3 - 3^n}{4} \]

6. **Combine the Terms**:
Combining both terms:
\[ S_n = \frac{(n - 1) \cdot 3^n}{2} + \frac{3 - 3^n}{4} \]
To combine these fractions, find a common denominator (which is 4):
\[ S_n = \frac{2(n - 1) \cdot 3^n + 3 - 3^n}{4} \]
Simplifying further:
\[ S_n = \frac{(2n - 2) \cdot 3^n + 3 - 3^n}{4} = \frac{(2n - 1) \cdot 3^n + 3}{4} \]

7. **Comparing with Given Equation**:
We have:
\[ S_n = \frac{(2n - 1) \cdot 3^n + b}{4} \]
By comparing both sides, we can see that:
- The coefficient of \( 3^n \) matches, confirming \( a = 1 \).
- The constant term gives \( b = 3 \).

### Final Values:
Thus, the values of \( a \) and \( b \) are:
\[ a = 1, \quad b = 3 \]