Question

Find the 9th term of the arithmetic sequence 4x+4, 9x+11, 14x+18

Answer 1

To find the 9th term of the arithmetic sequence given, we first need to determine the first term and the common difference of this sequence.

The first three terms of the sequence are:
- First term \( a_1 = 4x + 4 \)
- Second term \( a_2 = 9x + 11 \)
- Third term \( a_3 = 14x + 18 \)

**Step 1: Calculate the common difference**

The common difference \( d \) can be found by subtracting the first term from the second term:

\[
d = a_2 - a_1 = (9x + 11) - (4x + 4)
\]
\[
d = 9x + 11 - 4x - 4 = 5x + 7
\]

We can also check the difference between the third and second terms:

\[
d = a_3 - a_2 = (14x + 18) - (9x + 11)
\]
\[
d = 14x + 18 - 9x - 11 = 5x + 7
\]

Thus, the common difference \( d \) is confirmed to be \( 5x + 7 \).

**Step 2: Write the formula for the \( n \)-th term**

The \( n \)-th term of an arithmetic sequence can be found using the formula:

\[
a_n = a_1 + (n - 1)d
\]

Substituting for \( a_1 \) and \( d \):

\[
a_n = (4x + 4) + (n - 1)(5x + 7)
\]

**Step 3: Find the 9th term**

We need to find the 9th term \( a_9 \):

\[
a_9 = (4x + 4) + (9 - 1)(5x + 7)
\]
\[
= (4x + 4) + 8(5x + 7)
\]
\[
= (4x + 4) + (40x + 56)
\]
\[
= 4x + 4 + 40x + 56
\]
\[
= (4x + 40x) + (4 + 56)
\]
\[
= 44x + 60
\]

Thus, the 9th term of the arithmetic sequence is

\[
\boxed{44x + 60}
\]