Question

Find the 11th term of the arithmetic sequence x-7, 4x-1, 7x+5

Answer 1

To find the 11th term of the arithmetic sequence given by the first three terms \( x - 7 \), \( 4x - 1 \), and \( 7x + 5 \), we need to follow these steps:

1. **Determine the common difference:** In an arithmetic sequence, the common difference \( d \) is the difference between any two consecutive terms.

We first calculate the difference between the second term and the first term:

\[
d = (4x - 1) - (x - 7)
\]
Simplifying this:
\[
d = 4x - 1 - x + 7 = 3x + 6
\]

Now, we calculate the difference between the third term and the second term:

\[
d = (7x + 5) - (4x - 1)
\]
Simplifying this:
\[
d = 7x + 5 - 4x + 1 = 3x + 6
\]

Thus, the common difference \( d \) is \( 3x + 6 \).

2. **Find the first term \( a \):** The first term \( a \) is \( x - 7 \).

3. **Use the formula for the nth term of an arithmetic sequence:** The nth term \( a_n \) of an arithmetic sequence can be expressed as:

\[
a_n = a + (n-1) \cdot d
\]

For the 11th term (\( n = 11 \)):

\[
a_{11} = a + (11-1) \cdot d
\]
\[
a_{11} = (x - 7) + 10 \cdot (3x + 6)
\]

4. **Calculate \( a_{11} \):**

\[
a_{11} = (x - 7) + 10(3x + 6)
\]
\[
a_{11} = (x - 7) + (30x + 60)
\]
\[
a_{11} = x + 30x - 7 + 60
\]
\[
a_{11} = 31x + 53
\]

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

\[
\boxed{31x + 53}
\]