If the sum of two extreme numbers of an A.P with four terms is 8 and the product of remaining two middle terms is 15, then the greatest number will be

Options

A. 5
B. 7
C. 9
D. 11

Answer

To solve the problem step by step, we need to find the greatest number in an arithmetic progression (A.P) with four terms, given the conditions about the sum of the extreme terms and the product of the middle terms.

### Step 1: Define the terms of the A.P.
Let the four terms of the A.P be:
- First term: \( a - 3d \)
- Second term: \( a - d \)
- Third term: \( a + d \)
- Fourth term: \( a + 3d \)

### Step 2: Use the information about the sum of the extreme terms.
According to the problem, the sum of the two extreme terms is 8:
\[ (a - 3d) + (a + 3d) = 8 \]
This simplifies to:
\[ 2a = 8 \]
From this, we can find \( a \):
\[ a = \frac{8}{2} = 4 \]

### Step 3: Use the information about the product of the middle terms.
The product of the middle terms is given as 15:
\[ (a - d)(a + d) = 15 \]
Using the difference of squares, this can be rewritten as:
\[ a^2 - d^2 = 15 \]
Substituting \( a = 4 \) into the equation:
\[ 4^2 - d^2 = 15 \]
This simplifies to:
\[ 16 - d^2 = 15 \]
Rearranging gives:
\[ d^2 = 1 \]
Taking the square root, we find:
\[ d = 1 \quad \text{or} \quad d = -1 \]

### Step 4: Find the terms of the A.P.
Now we will find the terms of the A.P. using \( a = 4 \) and \( d = 1 \):
- First term: \( 4 - 3(1) = 1 \)
- Second term: \( 4 - 1 = 3 \)
- Third term: \( 4 + 1 = 5 \)
- Fourth term: \( 4 + 3(1) = 7 \)

Thus, the terms of the A.P are \( 1, 3, 5, 7 \).

### Step 5: Identify the greatest number.
The greatest number in this A.P is:
\[ \text{Greatest number} = 7 \]

### Final Answer:
The greatest number in the A.P is \( 7 \).

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