The sum of three decreasing numbers in A.P. is 27. If-1,-1, 3 are added to them respectively, the resulting series is in G.P. The numbers are

Options

A. 5,9,13
B. 15,9,3
C. 13,9,5
D. 17,9,1

Answer

To solve the problem, we need to find three decreasing numbers in Arithmetic Progression (A.P.) whose sum is 27. Additionally, when we add -1, -1, and 3 to these numbers respectively, the resulting numbers form a Geometric Progression (G.P.).

### Step-by-Step Solution:

1. **Define the A.P. Terms:**
Let the three numbers in A.P. be:
- First term: \( a + d \)
- Second term: \( a \)
- Third term: \( a - d \)

2. **Set Up the Equation for the Sum:**
According to the problem, the sum of these three terms is 27:
\[ (a + d) + a + (a - d) = 27 \]
Simplifying this, we have:
\[ 3a = 27 \]
Therefore, we find:
\[ a = 9 \]

3. **Substitute \( a \) Back into the A.P. Terms:**
Now substituting \( a = 9 \) into the terms, we have:
- First term: \( 9 + d \)
- Second term: \( 9 \)
- Third term: \( 9 - d \)

4. **Add the Given Values to the A.P. Terms:**
According to the problem, we add -1, -1, and 3 to the terms respectively:
- First term becomes: \( (9 + d) - 1 = 8 + d \)
- Second term becomes: \( 9 - 1 = 8 \)
- Third term becomes: \( (9 - d) + 3 = 12 - d \)

5. **Set Up the G.P. Condition:**
The three resulting terms \( 8 + d, 8, 12 - d \) must be in G.P. For three numbers to be in G.P., the square of the middle term must equal the product of the other two terms:
\[ 8^2 = (8 + d)(12 - d) \]
This simplifies to:
\[ 64 = (8 + d)(12 - d) \]

6. **Expand and Rearrange the Equation:**
Expanding the right side:
\[ 64 = 96 - 8d + 12d - d^2 \]
Rearranging gives:
\[ d^2 - 4d - 32 = 0 \]

7. **Solve the Quadratic Equation:**
We can factor this quadratic:
\[ (d - 8)(d + 4) = 0 \]
Thus, the solutions for \( d \) are:
\[ d = 8 \quad \text{or} \quad d = -4 \]

8. **Find the A.P. Terms for Each \( d \):**
- If \( d = 8 \):
- The terms are \( 9 + 8 = 17, 9, 9 - 8 = 1 \) (which are decreasing).
- If \( d = -4 \):
- The terms are \( 9 - 4 = 5, 9, 9 + 4 = 13 \) (which are increasing).

9. **Conclusion:**
Since we need decreasing numbers, the valid solution is:
\[ 17, 9, 1 \]

### Final Answer:
The three decreasing numbers in A.P. are **17, 9, and 1**.