Final answer:
By using the sum formula for arithmetic progression and solving a system of equations, we have determined that an error may exist as the derived first term and common difference do not match any of the given answer choices.
Step-by-step explanation:
In the given arithmetic progression, the sum of the first 10 terms is -150 and the sum of the next 10 terms is -550. The sum of an arithmetic sequence can be found using the formula: Sn = ½ n (2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Let's calculate the sum of the first 20 terms to find the relationship between the two sums given: S20 = S10 (first 10 terms) + S10 (next 10 terms) = -150 - 550 = -700
We have two equations now:
S10 = ½ * 10 (2a + 9d) = -150
S20 = ½ * 20 (2a + 19d) = -700
From the first equation, we can get 2a + 9d = -30. We can also rewrite the second equation to find another relation between a and d.
Dividing the second equation by 20 gives us:2a + 19d = -70. Now, we have a system of equations to solve:
2a + 9d = -30
2a + 19d = -70
Subtracting the first equation from the second gives us:10d = -40, which gives d = -4. Substitute d in the first equation to find a:
2a + 9(-4) = -30 → 2a - 36 = -30 → 2a = 6 → a = 3.