I see you’ve uploaded an image containing questions about arithmetic and geometric sequences. I’ll provide solutions for each part in detail:

1. Arithmetic Sequence

Determine the first term ($a$) and common difference ($b$):

• (a) 2, 5, 8, 11, ...

  • $a = 2$
  • $b = 5 - 2 = 3$

• (b) 3, 8, 13, 18, ...

  • $a = 3$
  • $b = 8 - 3 = 5$

• (c) 100, 90, 80, 70, ...

  • $a = 100$
  • $b = 90 - 100 = -10$

• (d) $4, 3\frac{1}{2}, 3, 2\frac{1}{2}, ...$

  • $a = 4$
  • $b = 3\frac{1}{2} - 4 = -\frac{1}{2}$

• (e) 4, 2, 0, -2, ...

  • $a = 4$
  • $b = 2 - 4 = -2$

2. Find the General Term ($U_n$):

The formula for the general term of an arithmetic sequence is: $U_n = a + (n - 1)b$

• (a) 2, 5, 8, 11, ...

  • $U_n = 2 + (n - 1)3 = 3n - 1$

• (b) 13, 18, 23, 28, ...

  • $U_n = 13 + (n - 1)5 = 5n + 8$

• (c) 100, 90, 80, 70, ...

  • $U_n = 100 + (n - 1)(-10) = 110 - 10n$

3. Find the Number of Terms ($n$):

Use the formula: $U_n = a + (n - 1)b$

• (a) $2, 4, 6, 8, ..., 80$:

  • $80 = 2 + (n - 1)2$
  • $80 - 2 = 2(n - 1)$
  • $78 = 2n - 2$
  • $80 = 2n$
  • $n = 40$

• (b) $30, 27, 24, ..., 6$:

  • $6 = 30 + (n - 1)(-3)$
  • $6 - 30 = -3(n - 1)$
  • $-24 = -3n + 3$
  • $-27 = -3n$
  • $n = 9$

4. Geometric Sequence

Determine the first term ($a$) and ratio ($r$):

The ratio ($r$) is determined by dividing consecutive terms ($\frac{U_2}{U_1}$).

• (a) $1, 4, 16, 64, ...$

  • $a = 1$
  • $r = \frac{4}{1} = 4$

• (b) $3, -6, 12, -24, ...$

  • $a = 3$
  • $r = \frac{-6}{3} = -2$

• (c) $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, ...$

  • $a = 1$
  • $r = \frac{\frac{1}{3}}{1} = \frac{1}{3}$

Do you need detailed explanations for each calculation?
Here are 5 additional practice questions you might want to try:

1. Write the first 5 terms of the arithmetic sequence with $a = 7$ and $b = 2$.
2. Derive the general term for the sequence $10, 15, 20, 25, ...$.
3. How many terms are there in the sequence $50, 45, 40, ..., 5$?
4. For the geometric sequence $2, 6, 18, 54, ...$, find the 7th term.
5. Determine the sum of the first 20 terms of the sequence $5, 10, 15, 20, ...$.

Tip: Always double-check calculations involving fractions or negative differences to avoid mistakes.