Certainly! Let's solve this step by step:
1. Identify the pattern in James' typing speed:
- At the end of the first month: 9 words per minute.
- At the end of the second month: 18 words per minute.
- At the end of the third month: 27 words per minute.
Observing these values, we can see that James' typing speed forms an arithmetic sequence (each term increases by the same amount).
2. Determine the common difference:
- From the first month to the second month: [tex]\(18 - 9 = 9\)[/tex]
- From the second month to the third month: [tex]\(27 - 18 = 9\)[/tex]
So, the common difference (denoted as [tex]\(d\)[/tex]) is 9 words per minute.
3. Formulate the general term of the arithmetic sequence:
The general formula for the n-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
where [tex]\(a_n\)[/tex] is the n-th term, [tex]\(a_1\)[/tex] is the first term, and [tex]\(d\)[/tex] is the common difference.
4. Apply the formula to find James' typing speed at the end of the fifth month:
- First term ([tex]\(a_1\)[/tex]) = 9 words per minute.
- Common difference ([tex]\(d\)[/tex]) = 9 words per minute.
- n = 5 (since we are asked about the fifth month).
Plugging these values into the formula gives us:
[tex]\[
a_5 = 9 + (5 - 1) \times 9
\][/tex]
Simplifying inside the parentheses:
[tex]\[
a_5 = 9 + 4 \times 9
\][/tex]
Multiply:
[tex]\[
a_5 = 9 + 36
\][/tex]
Adding these together:
[tex]\[
a_5 = 45
\][/tex]
Therefore, at the end of five months, James could type 45 words per minute.
Thus, the correct answer is:
b. 45 words per minute.
