Question

5,8,11,14,…
let’s use a table to write an equation for this sequence.
the first three rows have been completed for you. use them as an example to help you complete the table.
this message will change when you’ve correctly filled out the table.
term number (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | … | 19 | … | n
work | 5 | 5 + 3 | 5 + 3 + 3 | | | | | … | | … |
simplified expression | 5 | 5 + 3·1 | 5 + 3·2 | | | | | … | | … |
term (f(n)) | 5 | 8 | 11 | 14 | | | | … | | … | f(n)

Explanation:

Step1: Identify the pattern

Looking at the simplified expressions: for \( n = 1 \), it's \( 5 \); for \( n = 2 \), \( 5 + 3\cdot1 \); for \( n = 3 \), \( 5 + 3\cdot2 \). So the pattern is \( f(n)=5 + 3(n - 1) \). Simplify this: \( f(n)=3n + 2 \).

Step2: Fill in for \( n = 4 \)

Work: \( 5+3+3+3 \) (since we add 3 three times for \( n = 4 \), as \( n - 1 = 3 \)). Simplified Expression: \( 5 + 3\cdot3 \). Term: \( 3(4)+2 = 14 \) (matches given).

Step3: Fill in for \( n = 5 \)

Work: \( 5+3+3+3+3 \) (add 3 four times). Simplified Expression: \( 5 + 3\cdot4 \). Term: \( 3(5)+2 = 17 \).

Step4: Fill in for \( n = 6 \)

Work: \( 5+3+3+3+3+3 \) (add 3 five times). Simplified Expression: \( 5 + 3\cdot5 \). Term: \( 3(6)+2 = 20 \).

Step5: Fill in for \( n = 7 \)

Work: \( 5+3+3+3+3+3+3 \) (add 3 six times). Simplified Expression: \( 5 + 3\cdot6 \). Term: \( 3(7)+2 = 23 \).

Step6: Fill in for \( n = 19 \)

Simplified Expression: \( 5 + 3\cdot18 \) (since \( n - 1 = 18 \)). Term: \( 3(19)+2 = 59 \).

Answer:

The general formula is \( f(n)=3n + 2 \), and the table can be filled as follows:

Term Number (\( n \))	Work	Simplified Expression	Term (\( f(n) \))
2	5+3	\( 5 + 3\cdot1 \)	8
3	5+3+3	\( 5 + 3\cdot2 \)	11
4	5+3+3+3	\( 5 + 3\cdot3 \)	14
5	5+3+3+3+3	\( 5 + 3\cdot4 \)	17
6	5+3+3+3+3+3	\( 5 + 3\cdot5 \)	20
7	5+3+3+3+3+3+3	\( 5 + 3\cdot6 \)	23
...	...	...	...
19	-	\( 5 + 3\cdot18 \)	59
\( n \)	-	\( 5 + 3(n - 1) \) or \( 3n + 2 \)	\( 3n + 2 \)