To find the $N$th term of each sequence, we first need to identify the pattern in the sequence, which is usually an arithmetic progression (AP) or geometric progression (GP). Here are the steps for each sequence:

1. Sequence: 1, 5, 9, 13, ...

• Type: Arithmetic Progression (AP)
• First Term ( $a$ ): 1
• Common Difference ( $d$ ): 4
• Formula for the $N$th term: $a_N = a + (N-1) \times d$
• 27th Term: $a_{27} = 1 + (27-1) \times 4 = 1 + 26 \times 4 = 1 + 104 = 105$

2. Sequence: 20, 15, 10, ...

• Type: Arithmetic Progression (AP)
• First Term ( $a$ ): 20
• Common Difference ( $d$ ): -5
• Formula for the $N$th term: $a_N = a + (N-1) \times d$
• 66th Term: $a_{66} = 20 + (66-1) \times (-5) = 20 + 65 \times (-5) = 20 - 325 = -305$

3. Sequence: 100, 92, 84, ...

• Type: Arithmetic Progression (AP)
• First Term ( $a$ ): 100
• Common Difference ( $d$ ): -8
• Formula for the $N$th term: $a_N = a + (N-1) \times d$
• 48th Term: $a_{48} = 100 + (48-1) \times (-8) = 100 + 47 \times (-8) = 100 - 376 = -276$

4. Sequence: 7, 14, 21, ...

• Type: Arithmetic Progression (AP)
• First Term ( $a$ ): 7
• Common Difference ( $d$ ): 7
• Formula for the $N$th term: $a_N = a + (N-1) \times d$
• 89th Term: $a_{89} = 7 + (89-1) \times 7 = 7 + 88 \times 7 = 7 + 616 = 623$

5. Sequence: 1, $\frac{1}{2}$, 0, ...

• Type: Arithmetic Progression (AP)
• First Term ( $a$ ): 1
• Common Difference ( $d$ ): $\frac{-1}{2}$
• Formula for the $N$th term: $a_N = a + (N-1) \times d$
• 16th Term: $a_{16} = 1 + (16-1) \times \left(\frac{-1}{2}\right) = 1 + 15 \times \left(\frac{-1}{2}\right) = 1 - \frac{15}{2} = 1 - 7.5 = -6.5$

Would you like more details or have any questions? Here are five questions to consider:

1. How do you determine if a sequence is arithmetic or geometric?
2. Can you derive the formula for the sum of the first $N$ terms in an arithmetic sequence?
3. What would be the 50th term in the sequence 20, 15, 10,...?
4. How does the common difference affect the sequence's behavior over many terms?
5. How would you find the Nth term for a geometric sequence?

Tip: For arithmetic sequences, knowing the first term and the common difference allows you to quickly determine any term in the sequence!