To solve this question, let's tackle each arithmetic progression (AP) separately.

Part (a): AP = 1, 8, 15, ..., 78

1. Find the Common Difference (d): The common difference $d$ can be found by subtracting the first term from the second term: $d = 8 - 1 = 7$

2. Write the General Formula for the $n$th Term: The formula for the $n$th term of an AP is: $a_n = a + (n - 1) \cdot d$ where $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

   Substituting $a = 1$ and $d = 7$: $a_n = 1 + (n - 1) \cdot 7 = 1 + 7n - 7 = 7n - 6$

3. Determine the Number of Terms (n) Needed to Reach 78: Set $a_n = 78$ and solve for $n$: $7n - 6 = 78$ $7n = 84$ $n = 12$

   So, there are 12 terms in this AP, and the $n$th term formula is $a_n = 7n - 6$.

Part (b): AP = $\frac{1}{2}$, $\frac{3}{4}$, $1$, ..., $\frac{11}{2}$

1. Find the Common Difference (d): Subtract the first term from the second term: $d = \frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$

2. Write the General Formula for the $n$th Term: Using $a = \frac{1}{2}$ and $d = \frac{1}{4}$: $a_n = a + (n - 1) \cdot d$ $a_n = \frac{1}{2} + (n - 1) \cdot \frac{1}{4}$ $= \frac{1}{2} + \frac{n - 1}{4} = \frac{2 + n - 1}{4} = \frac{n + 1}{4}$

3. Determine the Number of Terms (n) Needed to Reach $\frac{11}{2}$: Set $a_n = \frac{11}{2}$ and solve for $n$: $\frac{n + 1}{4} = \frac{11}{2}$ Multiply both sides by 4: $n + 1 = 22$ $n = 21$

   So, there are 21 terms in this AP, and the $n$th term formula is $a_n = \frac{n + 1}{4}$.

Summary:

• For Part (a): The number of terms is 12, and the $n$th term formula is $a_n = 7n - 6$.
• For Part (b): The number of terms is 21, and the $n$th term formula is $a_n = \frac{n + 1}{4}$.

Would you like more details or have any questions about these steps?

Related Questions

1. How do you find the common difference in an arithmetic sequence?
2. What is the formula for the sum of the first $n$ terms in an AP?
3. How can you determine if a sequence is arithmetic?
4. How would the formula for the $n$th term change if the common difference was negative?
5. How can you use the $n$th term formula to find any specific term in an AP?

Tip

When solving for the $n$th term in an AP, always remember to start by identifying the first term and common difference accurately; they are crucial to deriving the correct formula.