The sum of the first 25 terms of an arithmetic sequence is 1,400, and the 25th term is 104. If the first term of the sequence is `a_(1)` and the second term is `a_(2)`, what is the value of `a_(2)-a_(1)` ?

Options

A. `-3`
B. 2
C. 4
D. 5

Explanation

To solve the problem step by step, we will use the properties of an arithmetic sequence.

### Step 1: Use the formula for the sum of the first n terms of an arithmetic sequence.
The formula for the sum of the first n terms (S_n) of an arithmetic sequence is given by:

\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]

where \(a_1\) is the first term, \(a_n\) is the nth term, and \(n\) is the number of terms.

### Step 2: Substitute the known values into the formula.
We know that:
- \(S_{25} = 1400\)
- \(n = 25\)
- \(a_{25} = 104\)

Substituting these values into the formula gives:

\[ 1400 = \frac{25}{2} \times (a_1 + 104) \]

### Step 3: Solve for \(a_1\).
To isolate \(a_1\), first multiply both sides by 2 to eliminate the fraction:

\[ 2800 = 25 \times (a_1 + 104) \]

Now, divide both sides by 25:

\[ 112 = a_1 + 104 \]

Now, subtract 104 from both sides:

\[ a_1 = 112 - 104 = 8 \]

### Step 4: Use the formula for the nth term to find the common difference \(d\).
The formula for the nth term of an arithmetic sequence is:

\[ a_n = a_1 + (n - 1) \cdot d \]

For \(n = 25\):

\[ a_{25} = a_1 + 24d \]

Substituting the known values:

\[ 104 = 8 + 24d \]

### Step 5: Solve for \(d\).
Subtract 8 from both sides:

\[ 96 = 24d \]

Now, divide both sides by 24:

\[ d = \frac{96}{24} = 4 \]

### Step 6: Find \(a_2 - a_1\).
The second term \(a_2\) can be expressed as:

\[ a_2 = a_1 + d \]

Thus,

\[ a_2 - a_1 = d \]

Since we found \(d = 4\):

\[ a_2 - a_1 = 4 \]

### Final Answer:
The value of \(a_2 - a_1\) is \(4\).

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