Let's analyze each part of the problem:

Part (a)

The sequence is: $1, 8, 15, \dots, 78$.

1. Identify the first term ($a$) and the common difference ($d$):

   • First term, $a = 1$.
   • Common difference, $d = 8 - 1 = 7$.

2. General formula for the $n$-th term in an Arithmetic Progression (AP): $a_n = a + (n - 1)d$ Substituting $a = 1$ and $d = 7$: $a_n = 1 + (n - 1) \cdot 7$ Simplifying, $a_n = 1 + 7n - 7 = 7n - 6$ So, the $n$-th term is $a_n = 7n - 6$.

3. Find the number of terms: Since the last term is 78, we can set $a_n = 78$ and solve for $n$: $7n - 6 = 78$ $7n = 84$ $n = 12$ Therefore, there are 12 terms in this sequence.

Part (b)

The sequence is: $\frac{1}{2}, \frac{3}{4}, 1, \dots, 5\frac{1}{2}$.

1. Identify the first term ($a$) and the common difference ($d$):

   • First term, $a = \frac{1}{2}$.
   • Common difference, $d = \frac{3}{4} - \frac{1}{2} = \frac{3 - 2}{4} = \frac{1}{4}$.

2. General formula for the $n$-th term in an Arithmetic Progression (AP): $a_n = a + (n - 1)d$ Substituting $a = \frac{1}{2}$ and $d = \frac{1}{4}$: $a_n = \frac{1}{2} + (n - 1) \cdot \frac{1}{4}$ Simplifying, $a_n = \frac{1}{2} + \frac{n - 1}{4} = \frac{2 + n - 1}{4} = \frac{n + 1}{4}$ So, the $n$-th term is $a_n = \frac{n + 1}{4}$.

3. Find the number of terms: Since the last term is $5\frac{1}{2} = \frac{11}{2}$, we set $a_n = \frac{11}{2}$ and solve for $n$: $\frac{n + 1}{4} = \frac{11}{2}$ Cross-multiplying, $2(n + 1) = 44$ $2n + 2 = 44$ $2n = 42$ $n = 21$ Therefore, there are 21 terms in this sequence.

Summary

• Part (a):

  • Number of terms = 12
  • $n$-th term = $7n - 6$

• Part (b):

  • Number of terms = 21
  • $n$-th term = $\frac{n + 1}{4}$

Would you like further details on any of these steps or have any questions?

Here are some related questions:

1. How do we derive the formula for the $n$-th term of an AP?
2. What would change if the common difference in an AP is negative?
3. How can we verify that a sequence is an AP?
4. How do we find the sum of terms in an AP?
5. How do we handle sequences where the first term or last term is missing?

Tip: When solving for the number of terms in an AP, always set up an equation with the last term and solve for $n$ using the general term formula.