Question

Find the 11th term of the arithmetic sequence minus, 4, x, plus, 2−4x+2, minus, 9, x, minus, 3−9x−3, minus, 14, x, minus, 8, comma, dot dot dot−14x−8,...

Answer 1

To find the 11th term of the given arithmetic sequence, we first need to identify the first term and the common difference.

The sequence starts with the terms:
- First term: \(-4\)
- Second term: \(-4x + 2\)
- Third term: \(-9x - 3\)
- Fourth term: \(-14x - 8\)

Next, we need to find the common difference between consecutive terms.

1. **Finding the second term minus the first term (common difference)**:
\[
d_1 = (-4x + 2) - (-4) = -4x + 2 + 4 = -4x + 6
\]

2. **Finding the third term minus the second term**:
\[
d_2 = (-9x - 3) - (-4x + 2) = -9x - 3 + 4x - 2 = -5x - 5
\]

3. **Finding the fourth term minus the third term**:
\[
d_3 = (-14x - 8) - (-9x - 3) = -14x - 8 + 9x + 3 = -5x - 5
\]

Now, notice that the common differences we've calculated show that the common difference appears to be \(-5x - 5\) starting from the second difference.

Now, with the first term \(a_1 = -4\) and the common difference \(d = -5x - 5\), we can express the \(n\)-th term of an arithmetic sequence as:
\[
a_n = a_1 + (n-1) d
\]

We are tasked with finding the 11th term (\(n = 11\)):
\[
a_{11} = -4 + (11-1)(-5x - 5)
\]
This simplifies to:
\[
a_{11} = -4 + 10(-5x - 5)
\]
Calculating inside the parenthesis:
\[
10(-5x - 5) = -50x - 50
\]
Now substituting back into the expression for \(a_{11}\):
\[
a_{11} = -4 - 50x - 50
\]
Combine the constants:
\[
a_{11} = -54 - 50x
\]

Thus, the 11th term of the arithmetic sequence is:
\[
\boxed{-50x - 54}
\]