Sure! Let’s solve Question 6 step by step.

Part (a):

First term: 9, term-to-term rule: add 9

1. Find the first six terms:
   Start with 9 and add 9 repeatedly:

   • 1st term: $9$
   • 2nd term: $9 + 9 = 18$
   • 3rd term: $18 + 9 = 27$
   • 4th term: $27 + 9 = 36$
   • 5th term: $36 + 9 = 45$
   • 6th term: $45 + 9 = 54$

   So, the first six terms are:
   9, 18, 27, 36, 45, 54

2. Position-to-term rule (nth term):
   Since the sequence increases by 9 each time, the formula is:
   $T_n = 9 + (n-1) \times 9$
   Simplify:
   $T_n = 9n$
   (This is the position-to-term rule where $T_n$ is the nth term.)

3. Find the 50th term:
   Using $T_n = 9n$:
   $T_{50} = 9 \times 50 = 450$

Part (b):

First term: 11, term-to-term rule: add 11

1. Find the first six terms:
   Start with 11 and add 11 repeatedly:

   • 1st term: $11$
   • 2nd term: $11 + 11 = 22$
   • 3rd term: $22 + 11 = 33$
   • 4th term: $33 + 11 = 44$
   • 5th term: $44 + 11 = 55$
   • 6th term: $55 + 11 = 66$

   So, the first six terms are:
   11, 22, 33, 44, 55, 66

2. Position-to-term rule (nth term):
   Since the sequence increases by 11 each time, the formula is:
   $T_n = 11 + (n-1) \times 11$
   Simplify:
   $T_n = 11n$
   (This is the position-to-term rule.)

3. Find the 50th term:
   Using $T_n = 11n$:
   $T_{50} = 11 \times 50 = 550$

Final Answers:

1. (a)

   • First six terms: 9, 18, 27, 36, 45, 54
   • Position-to-term rule: $T_n = 9n$
   • 50th term: 450

2. (b)

   • First six terms: 11, 22, 33, 44, 55, 66
   • Position-to-term rule: $T_n = 11n$
   • 50th term: 550

Let me know if you'd like a more detailed explanation or have any further questions!

Related Questions:

1. How do you derive the formula for an arithmetic sequence?
2. What is the general formula for the nth term of an arithmetic sequence?
3. How can you identify the common difference in a sequence?
4. What is the sum of the first $n$ terms in an arithmetic sequence?
5. How do you calculate the 100th term of a sequence with a common difference?

Tip: In arithmetic sequences, the common difference determines how much each term increases or decreases. The formula $T_n = a + (n-1)d$ always works!