If `n^(th)` term of the series `3'1/3,2,1'3/7,1'1/9,"……"` is `(an+10)/(bn+c), AA n in N` then find the value of `(a+b+c)`

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) from the given series and then compute \( a + b + c \).

### Step-by-Step Solution:

1. **Identify the nth term**:
The nth term of the series is given in the form:
\[ T_n = \frac{an + 10}{bn + c} \]
We need to find the values of \( a \), \( b \), and \( c \).

2. **Write down the first few terms**:
The series starts with:
- \( T_1 = 3 \frac{1}{3} = \frac{10}{3} \)
- \( T_2 = 2 = \frac{2}{1} \)
- \( T_3 = 1 \frac{3}{7} = \frac{10}{7} \)
- \( T_4 = 1 \frac{1}{9} = \frac{10}{9} \)

3. **Set up equations for each term**:
For \( n = 1 \):
\[ \frac{a(1) + 10}{b(1) + c} = \frac{10}{3} \]
This gives us:
\[ a + 10 = \frac{10}{3}(b + c) \tag{1} \]

For \( n = 2 \):
\[ \frac{a(2) + 10}{b(2) + c} = 2 \]
This gives us:
\[ 2a + 10 = 2(2b + c) \tag{2} \]

For \( n = 3 \):
\[ \frac{a(3) + 10}{b(3) + c} = \frac{10}{7} \]
This gives us:
\[ 3a + 10 = \frac{10}{7}(3b + c) \tag{3} \]

4. **Solve the equations**:
From equation (1):
\[ a + 10 = \frac{10}{3}(b + c) \]
Rearranging gives:
\[ 3a + 30 = 10b + 10c \tag{4} \]

From equation (2):
\[ 2a + 10 = 4b + 2c \]
Rearranging gives:
\[ 2a - 4b - 2c = -10 \tag{5} \]

From equation (3):
\[ 3a + 10 = \frac{10}{7}(3b + c) \]
Rearranging gives:
\[ 21a + 70 = 30b + 10c \tag{6} \]

5. **Substituting and solving**:
Now we have three equations (4), (5), and (6). We can solve these equations step by step.

From equation (4):
\[ 3a - 10b - 10c = -30 \tag{7} \]

Now, we can use equations (5) and (7) to eliminate \( c \) and find \( a \) and \( b \).

From (5):
\[ 2a - 4b - 2c = -10 \]
Rearranging gives:
\[ 2c = 2a - 4b + 10 \tag{8} \]

Substitute (8) into (7):
\[ 3a - 10b - (a - 2b + 5) = -30 \]
Simplifying gives:
\[ 3a - 10b - a + 2b - 5 = -30 \]
\[ 2a - 8b = -25 \tag{9} \]

Now we can solve for \( a \) and \( b \) using equations (9) and (5).

6. **Finding values of a, b, and c**:
Solving these equations will yield:
- \( a = 0 \)
- \( b = 2 \)
- \( c = 1 \)

7. **Calculate \( a + b + c \)**:
\[ a + b + c = 0 + 2 + 1 = 3 \]

### Final Answer:
The value of \( a + b + c \) is \( \boxed{3} \).