Sure! Let's work on finding the explicit formula for the arithmetic sequence step-by-step.
### Step 1: Identify the First Term
The first term of the sequence, [tex]\( a_1 \)[/tex], is provided as 17.
[tex]\[ a_1 = 17 \][/tex]
### Step 2: Determine the Common Difference (d)
To find the common difference [tex]\( d \)[/tex], we need the value of at least one more term. Here, we assume the second term, [tex]\( a_2 \)[/tex], is 20. Let's find [tex]\( d \)[/tex] by using the difference between the second term and the first term.
[tex]\[ d = a_2 - a_1 \][/tex]
[tex]\[ d = 20 - 17 \][/tex]
[tex]\[ d = 3 \][/tex]
### Step 3: Write the Explicit Formula
The explicit formula for an arithmetic sequence [tex]\( a_n \)[/tex] with the first term [tex]\( a_1 \)[/tex] and common difference [tex]\( d \)[/tex] is given by:
[tex]\[ a_n = a_1 + d(n-1) \][/tex]
Substitute the values of [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex] into the formula:
[tex]\[ a_n = 17 + 3(n-1) \][/tex]
### Step 4: Verify the Formula
Let’s verify the formula by calculating the values of the terms [tex]\( a_2 \)[/tex], [tex]\( a_3 \)[/tex], and [tex]\( a_4 \)[/tex].
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 17 + 3(2-1) \][/tex]
[tex]\[ a_2 = 17 + 3(1) \][/tex]
[tex]\[ a_2 = 17 + 3 \][/tex]
[tex]\[ a_2 = 20 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 17 + 3(3-1) \][/tex]
[tex]\[ a_3 = 17 + 3(2) \][/tex]
[tex]\[ a_3 = 17 + 6 \][/tex]
[tex]\[ a_3 = 23 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 17 + 3(4-1) \][/tex]
[tex]\[ a_4 = 17 + 3(3) \][/tex]
[tex]\[ a_4 = 17 + 9 \][/tex]
[tex]\[ a_4 = 26 \][/tex]
### Summary of the Term Values
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline \text{Term} \,( n )& 1 & 2 & 3 & 4 \\
\hline \text{Value} \,\left(a_n\right) & 17 & 20 & 23 & 26 \\
\hline
\end{array}
\][/tex]
### Final Explicit Formula
The explicit formula for the arithmetic sequence is:
[tex]\[ a_n = 17 + 3(n-1) \][/tex]
Thus, the complete explicit formula is provided and verified.
