Question

Find the 11th term of the arithmetic sequnce -x+5, 6x+1, 13x-3

Answer 1

To find the 11th term of the arithmetic sequence given by the terms \(-x + 5\), \(6x + 1\), and \(13x - 3\), we first need to determine the common difference of the sequence.

1. **Finding the common difference:**
Let's denote the first term as \(a_1 = -x + 5\), the second term as \(a_2 = 6x + 1\), and the third term as \(a_3 = 13x - 3\).

The common difference \(d\) can be calculated as:
\[
d = a_2 - a_1 = (6x + 1) - (-x + 5)
\]
Simplifying this gives:
\[
d = 6x + 1 + x - 5 = 7x - 4
\]

We can also calculate the difference between the third and second terms to confirm:
\[
d = a_3 - a_2 = (13x - 3) - (6x + 1)
\]
Simplifying this gives:
\[
d = 13x - 3 - 6x - 1 = 7x - 4
\]

Therefore, the common difference \(d\) is \(7x - 4\).

2. **Finding the 11th term:**
The \(n\)-th term of an arithmetic sequence is given by the formula:
\[
a_n = a_1 + (n-1)d
\]
For the 11th term (\(n = 11\)):
\[
a_{11} = a_1 + 10d
\]
Plugging in the values we calculated:
\[
a_{11} = (-x + 5) + 10(7x - 4)
\]
Now, simplify the equation:
\[
a_{11} = -x + 5 + 70x - 40
\]
Combine like terms:
\[
a_{11} = (-x + 70x) + (5 - 40) = 69x - 35
\]

Thus, the 11th term of the arithmetic sequence is:

\[
\boxed{69x - 35}
\]