Final answer:
The 31st term in the arithmetic sequence with the 22nd term as -143 and the 61st term as -416 is found to be -206, by determining the common difference and applying the arithmetic sequence formula.
Step-by-step explanation:
To find the 31st term in the sequence, given the 22nd term (-143) and the 61st term (-416), we assume it is an arithmetic sequence and use the formula for any term of an arithmetic sequence, which is an = a1 + (n-1)d, where a1 is the first term and d is the common difference.
We first need to find the common difference (d) using the given terms:
a61 = a22 + (61-22)d
So, -416 = -143 + 39d, and solving for d gives us d = (-416 + 143) / 39, which simplifies to d = -7.
Next, we can find the 31st term using the common difference:
a31 = a22 + (31-22)d
Plugging in the values, we get a31 = -143 + (9)(-7), and calculating this gives us a31 = -143 - 63, which equals -206. Thus, the 31st term of the sequence is -206.