If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then what is the sum of the first ten terms?

Options

A. `-500`
B. `-250`
C. 500
D. 250

Answer

To solve the problem step by step, we will follow the given conditions and use the formulas for the sum of an arithmetic progression (AP).

### Step 1: Identify the first term and the common difference
The first term of the AP is given as \( a = 2 \).

### Step 2: Write the formula for the sum of the first \( n \) terms of an AP
The sum of the first \( n \) terms \( S_n \) of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

### Step 3: Calculate the sum of the first 5 terms
Using the formula, we can find the sum of the first 5 terms \( S_5 \):
\[ S_5 = \frac{5}{2} \times (2 \times 2 + (5-1)d) = \frac{5}{2} \times (4 + 4d) = 5(2 + 2d) = 10 + 10d \]

### Step 4: Calculate the sum of the next 5 terms
The next 5 terms are the 6th to 10th terms. The sum of these terms can be expressed as:
\[ S_{10} - S_5 \]
where \( S_{10} \) is the sum of the first 10 terms. We can calculate \( S_{10} \) as follows:
\[ S_{10} = \frac{10}{2} \times (2 \times 2 + (10-1)d) = 5 \times (4 + 9d) = 20 + 45d \]
Thus, the sum of the next 5 terms is:
\[ S_{10} - S_5 = (20 + 45d) - (10 + 10d) = 10 + 35d \]

### Step 5: Set up the equation based on the problem statement
According to the problem, the sum of the first 5 terms is equal to one-fourth of the sum of the next 5 terms:
\[ S_5 = \frac{1}{4}(S_{10} - S_5) \]
Substituting the expressions we found:
\[ 10 + 10d = \frac{1}{4}(10 + 35d) \]

### Step 6: Solve the equation
Multiply both sides by 4 to eliminate the fraction:
\[ 4(10 + 10d) = 10 + 35d \]
This simplifies to:
\[ 40 + 40d = 10 + 35d \]
Rearranging gives:
\[ 40d - 35d = 10 - 40 \]
\[ 5d = -30 \]
\[ d = -6 \]

### Step 7: Calculate the sum of the first 10 terms
Now that we have \( d \), we can find \( S_{10} \):
\[ S_{10} = 20 + 45d = 20 + 45(-6) = 20 - 270 = -250 \]

### Final Answer
The sum of the first 10 terms is \( \boxed{-250} \).