What is the maximum sum of the terms in the arithmetic progression 25 , 24 1/2` , 24 ………?`
` A)`637 1/2` `
` B)625 `
` C)`662 1/2` `
` D)650 `
`

Options

A. `637 1/2`
B. 625
C. `662 1/2`
D. 650

Answer

To solve the problem of finding the maximum sum of the terms in the arithmetic progression (AP) given by 25, 24 1/2, 24, ..., we will follow these steps:

### Step 1: Identify the first term and common difference
- The first term \( a = 25 \).
- The second term is \( 24.5 \), which can be written as \( 24 \frac{1}{2} \).
- The common difference \( d \) can be calculated as:
\[ d = 24.5 - 25 = -0.5 \]

### Step 2: Determine the last term of the AP
- The AP is decreasing, and we need to find the last term before the terms become negative.
- The last term should be \( 0.5 \) (or \( \frac{1}{2} \)), as the next term would be \( 0 \), which does not contribute to the sum.

### Step 3: Use the formula for the nth term of an AP
- The formula for the nth term of an AP is given by:
\[ a_n = a + (n-1)d \]
- We set \( a_n = 0.5 \):
\[ 0.5 = 25 + (n-1)(-0.5) \]
- Rearranging gives:
\[ 0.5 - 25 = (n-1)(-0.5) \implies -24.5 = (n-1)(-0.5) \]
- Dividing both sides by \(-0.5\):
\[ n - 1 = \frac{24.5}{0.5} = 49 \implies n = 50 \]

### Step 4: Calculate the sum of the first n terms of the AP
- The formula for the sum \( S_n \) of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} (a + a_n) \]
- Substituting the values we found:
\[ S_{50} = \frac{50}{2} (25 + 0.5) = 25 \times 25.5 \]
- Calculating this gives:
\[ S_{50} = 25 \times 25.5 = 637.5 \]

### Step 5: Finalize the answer
- The maximum sum of the terms in the arithmetic progression is:
\[ 637 \frac{1}{2} \]

### Conclusion
The correct answer is option A: \( 637 \frac{1}{2} \).

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