What is the common difference of an arithmetic sequence if the sum of the second and fourth terms is 0, and the sum of the sixth and eighth terms is 12?

I'm really confused on this problem. If the sum of the second and fourth terms is 0 then the two terms cannot be 0 but a negative number that crosses each other out

first condition:

a+d + a+3d = 0
2a + 4d = 0
a + 2d = 0
a = -2d

a + 5d + a+ 7d = 12
2a + 12d = 12
a + 6d = 6

sub in the first part
-2d + 6d = 6
4d = 6
d = 3/2
then a = -2(3/2) = -3

check:
2nd term = -3+3/2 = -3/2
4th term = -3 + 3(3/2) = 3/2
their sum = -3/2 + 3/2 = 0

6th term = -3 + 5(3/2) = 9/2
8th term = -3 + 7(3/2) = 15/2
their sum = 9/2 + 15/2 = 12

Thanks a lot.

- Happy Holidays

To find the common difference of an arithmetic sequence, we need to analyze the given information.

Let's assume that the first term of the arithmetic sequence is 'a' and the common difference is 'd'.

The second term would then be 'a + d', the fourth term would be 'a + 3d', the sixth term would be 'a + 5d', and the eighth term would be 'a + 7d'.

According to the given information, the sum of the second and fourth terms is 0, so we can write the equation:

(a + d) + (a + 3d) = 0

By simplifying the equation, we have:

2a + 4d = 0

Similarly, the sum of the sixth and eighth terms is 12, so we can write the equation:

(a + 5d) + (a + 7d) = 12

Again, simplifying the equation, we have:

2a + 12d = 12

Now, we have a system of two linear equations with two variables:

2a + 4d = 0 ...(equation 1)
2a + 12d = 12 ...(equation 2)

To solve this system, we can use the method of substitution or elimination. I'll use the method of elimination to demonstrate:

By subtracting equation 1 from equation 2, we eliminate the 'a' term:

(2a + 12d) - (2a + 4d) = 12 - 0

The 'a' terms cancel out:

8d = 12

Dividing both sides by 8, we find:

d = 12 / 8
d = 3/2

Therefore, the common difference of the arithmetic sequence is 3/2 or 1.5.

To double-check our answer, we can substitute 'd' back into one of the original equations (equation 1 or 2) to find the value of 'a'. We can then verify that the second and fourth terms sum to 0, and the sixth and eighth terms sum to 12.