If the sum of n term of the series `(5)/(1.2.3) + (6)/(2.3.4) + ( 7)/(3.4.5)+"…."` is `a/2 - (n+b)/((n+1)(n+2))`, then

Options

A. `a = 3`
B. `b=2`
C. `b =3`
D. `a=2`

Answer

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

\[ S_n = \frac{5}{1 \cdot 2 \cdot 3} + \frac{6}{2 \cdot 3 \cdot 4} + \frac{7}{3 \cdot 4 \cdot 5} + \ldots + T_n \]

where \( S_n = \frac{a}{2} - \frac{n+b}{(n+1)(n+2)} \).

### Step 1: Identify the general term \( T_n \)

The general term \( T_n \) can be expressed as:

\[ T_n = \frac{n + 4}{n(n + 1)(n + 2)} \]

### Step 2: Use partial fractions to decompose \( T_n \)

We can express \( T_n \) in terms of partial fractions:

\[ \frac{n + 4}{n(n + 1)(n + 2)} = \frac{A}{n} + \frac{B}{n + 1} + \frac{C}{n + 2} \]

Multiplying through by \( n(n + 1)(n + 2) \) gives:

\[ n + 4 = A(n + 1)(n + 2) + Bn(n + 2) + Cn(n + 1) \]

### Step 3: Expand and compare coefficients

Expanding the right-hand side:

\[ A(n^2 + 3n + 2) + B(n^2 + 2n) + C(n^2 + n) \]

Combining like terms:

\[ (A + B + C)n^2 + (3A + 2B + C)n + 2A \]

Setting this equal to \( n + 4 \):

1. Coefficient of \( n^2 \): \( A + B + C = 0 \)
2. Coefficient of \( n \): \( 3A + 2B + C = 1 \)
3. Constant term: \( 2A = 4 \) → \( A = 2 \)

### Step 4: Solve for \( B \) and \( C \)

Substituting \( A = 2 \) into the equations:

1. \( 2 + B + C = 0 \) → \( B + C = -2 \)
2. \( 6 + 2B + C = 1 \) → \( 2B + C = -5 \)

Now we have a system of equations:

1. \( B + C = -2 \)
2. \( 2B + C = -5 \)

Subtracting the first from the second:

\[ (2B + C) - (B + C) = -5 + 2 \implies B = -3 \]

Substituting \( B = -3 \) into \( B + C = -2 \):

\[ -3 + C = -2 \implies C = 1 \]

### Step 5: Summation of the series

Now we have:

\[ A = 2, \quad B = -3, \quad C = 1 \]

Thus,

\[ T_n = \frac{2}{n} - \frac{3}{n + 1} + \frac{1}{n + 2} \]

### Step 6: Find the sum \( S_n \)

Now we can find the sum \( S_n \):

\[ S_n = \sum_{k=1}^{n} \left( \frac{2}{k} - \frac{3}{k + 1} + \frac{1}{k + 2} \right) \]

This can be computed by evaluating each term separately.

### Step 7: Analyze the resulting expression

After performing the summation, we can express \( S_n \) in the form given in the problem:

\[ S_n = \frac{3}{2} - \frac{n + 3}{(n + 1)(n + 2)} \]

### Step 8: Compare with the given expression

Comparing with \( S_n = \frac{a}{2} - \frac{n + b}{(n + 1)(n + 2)} \):

From our expression, we find:

- \( a = 3 \)
- \( b = 3 \)

### Final Answer

Thus, the values of \( a \) and \( b \) are:

\[ \boxed{3} \quad \text{and} \quad \boxed{3} \]