To solve the problem, we need to find the 4th term of the given arithmetic sequence. Let's go through the solution step-by-step.
### Step 1: Identify the given terms
The terms of the arithmetic sequence are given by:
[tex]\[ a_1 = 2x + 2 \][/tex]
[tex]\[ a_2 = 3x + 5 \][/tex]
[tex]\[ a_3 = 4x + 8 \][/tex]
[tex]\[ a_4 = 5 \][/tex]
### Step 2: Determine the common difference [tex]\( d \)[/tex]
The common difference [tex]\( d \)[/tex] in an arithmetic sequence is the difference between any two consecutive terms. We can find it as follows:
[tex]\[ d = a_2 - a_1 \][/tex]
Substituting the given terms:
[tex]\[ d = (3x + 5) - (2x + 2) \][/tex]
After simplifying:
[tex]\[ d = x + 3 \][/tex]
### Step 3: Find the nth term
The formula for the nth term of an arithmetic sequence [tex]\( a_n \)[/tex] is given by:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
Assuming we need to find the 4th term ([tex]\( n = 4 \)[/tex]):
[tex]\[ a_4 = a_1 + (4 - 1)d \][/tex]
Substituting the values of [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ a_4 = (2x + 2) + 3(x + 3) \][/tex]
### Step 4: Simplify the expression
[tex]\[ a_4 = (2x + 2) + 3(x + 3) \][/tex]
[tex]\[ a_4 = 2x + 2 + 3x + 9 \][/tex]
Combining like terms:
[tex]\[ a_4 = 5x + 11 \][/tex]
### Given Example
For a specific case where [tex]\( x = 1 \)[/tex]:
[tex]\[ a_4 = 5(1) + 11 \][/tex]
[tex]\[ a_4 = 5 + 11 \][/tex]
[tex]\[ a_4 = 16 \][/tex]
Thus, when [tex]\( x = 1 \)[/tex], the 4th term of the sequence is 16.
