Certainly! Let's solve this problem step-by-step:
We start with the given information:
- The first term of the arithmetic sequence ([tex]\( a_1 \)[/tex]) is 2.
- The sum of the first five terms ([tex]\( S_1 \)[/tex]) is equal to four times the sum of the next five terms ([tex]\( S_2 \)[/tex]).
### Step 1: Sum of the First Five Terms
An arithmetic sequence has the form [tex]\( a, a+d, a+2d, a+3d, \ldots \)[/tex]
For the first five terms:
[tex]\[ a_1, a_1 + d, a_1 + 2d, a_1 + 3d, a_1 + 4d \][/tex]
The sum of these terms ([tex]\( S_1 \)[/tex]) can be written as:
[tex]\[ S_1 = a_1 + (a_1 + d) + (a_1 + 2d) + (a_1 + 3d) + (a_1 + 4d) \][/tex]
Simplifying this:
[tex]\[ S_1 = 5a_1 + (0 + 1 + 2 + 3 + 4)d \][/tex]
[tex]\[ S_1 = 5a_1 + 10d \][/tex]
Given [tex]\( a_1 = 2 \)[/tex]:
[tex]\[ S_1 = 5(2) + 10d \][/tex]
[tex]\[ S_1 = 10 + 10d \][/tex]
### Step 2: Sum of the Next Five Terms
For the next five terms:
[tex]\[ a_1 + 5d, a_1 + 6d, a_1 + 7d, a_1 + 8d, a_1 + 9d \][/tex]
The sum of these terms ([tex]\( S_2 \)[/tex]) can be written as:
[tex]\[ S_2 = (a_1 + 5d) + (a_1 + 6d) + (a_1 + 7d) + (a_1 + 8d) + (a_1 + 9d) \][/tex]
Simplifying this:
[tex]\[ S_2 = 5a_1 + (5 + 6 + 7 + 8 + 9)d \][/tex]
[tex]\[ S_2 = 5a_1 + 35d \][/tex]
Given [tex]\( a_1 = 2 \)[/tex]:
[tex]\[ S_2 = 5(2) + 35d \][/tex]
[tex]\[ S_2 = 10 + 35d \][/tex]
### Step 3: Relate the Sums
According to the given problem:
[tex]\[ S_1 = 4S_2 \][/tex]
Substitute the expressions for [tex]\( S_1 \)[/tex] and [tex]\( S_2 \)[/tex]:
[tex]\[ 10 + 10d = 4(10 + 35d) \][/tex]
Simplify and solve for [tex]\( d \)[/tex]:
[tex]\[ 10 + 10d = 40 + 140d \][/tex]
Subtract 10 from both sides:
[tex]\[ 10d = 40 + 140d - 10 \][/tex]
[tex]\[ 10d = 30 + 140d \][/tex]
Subtract 10d from both sides:
[tex]\[ 0 = 30 + 130d \][/tex]
Subtract 30 from both sides:
[tex]\[ -30 = 130d \][/tex]
Divide by 130:
[tex]\[ d = \frac{-30}{130} \][/tex]
[tex]\[ d = -\frac{3}{13} \][/tex]
### Final Answer
The common difference [tex]\( d \)[/tex] of the arithmetic sequence is [tex]\( -\frac{3}{13} \)[/tex].
