To determine the values of x, the 20th term, and the sum of the first 20 terms of the arithmetic sequence, we'll need to use the given information about the first three terms.
(1) To find x:
We know that the common difference (d) between consecutive terms in an arithmetic sequence is constant. So, we can set up the following equations using the given terms:
Second term - First term = Third term - Second term
(2x + 6) - (x - 2) = (4x - 8) - (2x + 6)
Simplifying, we get:
x + 8 = 2x - 14
Rearranging and solving for x:
x - 2x = -14 - 8
- x = -22
x = 22
Therefore, x = 22.
(2) To find the 20th term:
In an arithmetic sequence, we can find the nth term (aā) using the formula:
aā = aā + (n - 1)d
In this case, we are given the first term (aā = x - 2), and the common difference (d = 2x + 6 - (x - 2) = x + 8). Plugging in these values:
aāā = (22 - 2) + (20 - 1)(22 + 8)
Simplifying, we get:
aāā = 20 + 19(30)
aāā = 20 + 570
aāā = 590
Therefore, the 20th term is 590.
(3) To find the sum of the first 20 terms:
The sum of the first n terms of an arithmetic sequence can be found using the formula:
Sn = (n/2)(2aā + (n - 1)d)
Plugging in the given values:
Sāā = (20/2)(2(x - 2) + (20 - 1)(x + 8))
Simplifying, we get:
Sāā = 10(2x - 4 + 19x + 152)
Sāā = 10(21x + 148)
Sāā = 210x + 1480
Therefore, the sum of the first 20 terms is 210x + 1480.