If `S _(r)` denote the sum of first 'r' terms of a non constaint A.P. and `(S_(a ))/(a ^(2)) =(S_(b))/(b ^(2))=c,` where a,b,c are distinct then `S_(c) =`

Options

A. `c ^(2)`
B. `c ^(3)`
C. `c ^(4)`
D. `abc`

Answer

To solve the problem, we need to find the value of \( S_c \) given the conditions regarding the sums of the first \( a \), \( b \), and \( c \) terms of a non-constant arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the Sum of the First \( n \) Terms of an A.P.**:
The sum of the first \( n \) terms \( S_n \) of an A.P. can be expressed as:
\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Expressing \( S_a \) and \( S_b \)**:
Using the formula for \( S_n \):
\[ S_a = \frac{a}{2} \left(2k + (a-1)d\right) \quad \text{(where \( k \) is the first term)} \]
\[ S_b = \frac{b}{2} \left(2k + (b-1)d\right) \]

3. **Setting Up the Given Condition**:
We have the conditions:
\[ \frac{S_a}{a^2} = \frac{S_b}{b^2} = c \]
This leads to:
\[ \frac{\frac{a}{2} \left(2k + (a-1)d\right)}{a^2} = c \]
\[ \frac{\frac{b}{2} \left(2k + (b-1)d\right)}{b^2} = c \]

4. **Simplifying the Equations**:
From the first equation:
\[ \frac{1}{2} \left(2k + (a-1)d\right) = \frac{c a^2}{a} = 2c \]
Thus:
\[ 2k + (a-1)d = 4c \quad \text{(1)} \]

From the second equation:
\[ \frac{1}{2} \left(2k + (b-1)d\right) = 2c \]
Thus:
\[ 2k + (b-1)d = 4c \quad \text{(2)} \]

5. **Setting the Two Equations Equal**:
Equating equations (1) and (2):
\[ 2k + (a-1)d = 2k + (b-1)d \]
This simplifies to:
\[ (a-1)d = (b-1)d \]
Since \( a \) and \( b \) are distinct, we can conclude that:
\[ d = 2k \]

6. **Finding \( S_c \)**:
Now, we can express \( S_c \):
\[ S_c = \frac{c}{2} \left(2k + (c-1)d\right) \]
Substituting \( d = 2k \):
\[ S_c = \frac{c}{2} \left(2k + (c-1)(2k)\right) = \frac{c}{2} \left(2k + 2k(c-1)\right) \]
Simplifying further:
\[ S_c = \frac{c}{2} \cdot 2kc = c^2 k \]

7. **Finding the Value of \( k \)**:
From the earlier equations, we substitute \( k = c \):
\[ S_c = c^2 \cdot c = c^3 \]

### Final Answer:
\[ S_c = c^3 \]