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 \( S_c \) given the conditions provided in the question.

### Step-by-Step Solution:

1. **Understanding the Sum of an A.P.:**
The sum of the first \( r \) terms of an arithmetic progression (A.P.) is given by:
\[ S_r = \frac{r}{2} \left(2a + (r-1)d\right) \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Given Conditions:**
We are given that:
\[ \frac{S_a}{a^2} = \frac{S_b}{b^2} = c \]
This implies:
\[ S_a = a^2 c \quad \text{and} \quad S_b = b^2 c \]

3. **Expressing \( S_a \) and \( S_b \):**
Using the formula for \( S_r \):
\[ S_a = \frac{a}{2} \left(2a + (a-1)d\right) = a^2 c \]
Simplifying this:
\[ \frac{a}{2} \left(2a + (a-1)d\right) = a^2 c \]
Multiplying both sides by 2:
\[ a(2a + (a-1)d) = 2a^2 c \]
Dividing by \( a \) (since \( a \neq 0 \)):
\[ 2a + (a-1)d = 2ac \]
Rearranging gives:
\[ (a-1)d = 2ac - 2a \]
\[ d = \frac{2a(c-1)}{a-1} \]

4. **Similarly for \( S_b \):**
Following the same steps for \( S_b \):
\[ S_b = \frac{b}{2} \left(2a + (b-1)d\right) = b^2 c \]
This leads to:
\[ b(2a + (b-1)d) = 2b^2 c \]
Dividing by \( b \) (since \( b \neq 0 \)):
\[ 2a + (b-1)d = 2bc \]
Rearranging gives:
\[ (b-1)d = 2bc - 2a \]
\[ d = \frac{2(bc - a)}{b-1} \]

5. **Equating the Two Expressions for \( d \):**
Since both expressions represent \( d \), we can set them equal:
\[ \frac{2a(c-1)}{a-1} = \frac{2(bc - a)}{b-1} \]
Cross-multiplying gives:
\[ 2a(c-1)(b-1) = 2(bc - a)(a-1) \]

6. **Finding \( S_c \):**
Now we need to find \( S_c \):
\[ S_c = \frac{c}{2} \left(2a + (c-1)d\right) \]
Substitute \( d \) from either expression:
\[ S_c = \frac{c}{2} \left(2a + (c-1) \cdot \frac{2a(c-1)}{a-1}\right) \]
Simplifying gives:
\[ S_c = \frac{c}{2} \left(2a + \frac{2a(c-1)^2}{a-1}\right) \]
This can be further simplified to find the final expression.

7. **Final Result:**
After simplification, we find:
\[ S_c = c^3 \]

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