Sure, let's find the 85th term of the arithmetic sequence 3, 7, 11, 15, 19, ...
### Step-by-Step Solution
1. Identify the first term ([tex]\(a_1\)[/tex]) of the sequence:
The first term of the sequence is given as [tex]\(3\)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]):
The common difference is the difference between two successive terms. For this sequence:
[tex]\[
d = 7 - 3 = 4
\][/tex]
3. Identify the term number ([tex]\(n\)[/tex]) you need to find:
We need to find the 85th term of the sequence, so [tex]\(n = 85\)[/tex].
4. Use the formula for the nth term ([tex]\(a_n\)[/tex]) of an arithmetic sequence:
The formula for the nth term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
5. Substitute the known values into the formula:
[tex]\[
a_{85} = 3 + (85 - 1) \cdot 4
\][/tex]
6. Simplify the expression:
[tex]\[
a_{85} = 3 + 84 \cdot 4
\][/tex]
[tex]\[
a_{85} = 3 + 336
\][/tex]
[tex]\[
a_{85} = 339
\][/tex]
Therefore, the 85th term of the arithmetic sequence is [tex]\(339\)[/tex].
