For a given arithmetic sequence, the 90th term, is equal to -230 , and the 8th term, is equal to 16 .
Find the value of the 31st term .
322
To find the value of the 31st term of the arithmetic sequence, we need to first find the common difference (d) of the sequence.
We are given that the 90th term is -230 and the 8th term is 16.
The formula to find the nth term of an arithmetic sequence is given by:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.
We can use this formula to form two equations:
1) For the 90th term:
-230 = a1 + (90 - 1)d
2) For the 8th term:
16 = a1 + (8 - 1)d
We now have a system of equations that we can solve to find the values of a1 and d.
Solving equation 2) for a1:
a1 = 16 - 7d
Substituting this value of a1 in equation 1):
-230 = (16 - 7d) + (90 - 1)d
-230 = 16 - 7d + 90d - d
-230 = 16 + 89d
Rearranging the equation:
89d = -230 - 16
89d = -246
d = -246/89
d ≈ -2.764
Now that we have the value of the common difference (d), we can find the first term (a1) by substituting the value of d into equation 2):
a1 = 16 - 7(-2.764)
a1 = 16 + 19.348
a1 ≈ 35.348
Finally, we can find the value of the 31st term (a31) using the formula:
a31 = a1 + (31 - 1)d
a31 = 35.348 + (31 - 1)(-2.764)
a31 ≈ 35.348 + 29(-2.764)
a31 ≈ 35.348 - 80.156
a31 ≈ -44.808
Therefore, the value of the 31st term is approximately -44.808.