Let's address each part of the exercise one by one:
(a) State the rule in words for this sequence:
Looking at the given sequence: 2, 7, 12, 17, we notice that each term increases by 5 from the previous one. Hence, the rule in words for this sequence is: "Start at 2 and add 5 to get the next term."
(1) Continue the pattern for the next four terms:
To continue the pattern, we will just keep adding 5 to the last known term.
- Term 5: \( 17 + 5 = 22 \)
- Term 6: \( 22 + 5 = 27 \)
- Term 7: \( 27 + 5 = 32 \)
- Term 8: \( 32 + 5 = 37 \)
So the next four terms are 22, 27, 32, and 37.
What is the 9th term?
To find the 9th term, we continue the pattern:
- Term 9: \( 37 + 5 = 42 \)
Therefore, the 9th term is 42.
Which term of the sequence is 57?
To find which term is 57, we set up an equation with what is known from the rule:
\[ T_n = 2 + 5(n-1) \]
Now we solve for n when \( T_n = 57 \):
\[ 57 = 2 + 5(n - 1) \]
Subtract 2 from both sides:
\[ 55 = 5(n - 1) \]
Divide both sides by 5:
\[ 11 = n - 1 \]
Add 1 to both sides:
\[ n = 12 \]
So the term that is 57 is the 12th term.
Determine the 14th term:
To find the 14th term, we will use the rule again:
\[ T_{14} = 2 + 5(14 - 1) \]
\[ T_{14} = 2 + 5(13) \]
\[ T_{14} = 2 + 65 \]
\[ T_{14} = 67 \]
The 14th term of the sequence is 67.
