Let's analyze the problem step by step.

Part (a): Geometric Sequence

We are given:

• The 4th term ($a_4$) = 24
• The 7th term ($a_7$) = 192

General formula for a geometric sequence:

$a_n = a \cdot r^{n-1}$ where:

• $a$ = first term
• $r$ = common ratio

From the formula: $a_4 = a \cdot r^3 = 24 \quad \text{(1)}$ $a_7 = a \cdot r^6 = 192 \quad \text{(2)}$

Solve for $r$:

Dividing equation (2) by equation (1): $\frac{a \cdot r^6}{a \cdot r^3} = \frac{192}{24}$ $r^3 = 8 \quad \implies \quad r = 2$

Solve for $a$:

Substitute $r = 2$ into equation (1): $a \cdot 2^3 = 24$ $a \cdot 8 = 24 \quad \implies \quad a = 3$

Final expression for $a_n$:

$a_n = 3 \cdot 2^{n-1}$

Part (c): Arithmetic Sequence

We are given:

• The 6th term ($a_6$) = 13
• The 14th term ($a_{14}$) = 33

General formula for an arithmetic sequence:

$a_n = a + (n-1)d$ where:

• $a$ = first term
• $d$ = common difference

From the formula: $a_6 = a + 5d = 13 \quad \text{(1)}$ $a_{14} = a + 13d = 33 \quad \text{(2)}$

Solve for $d$:

Subtract equation (1) from equation (2): $(a + 13d) - (a + 5d) = 33 - 13$ $8d = 20 \quad \implies \quad d = 2.5$

Solve for $a$:

Substitute $d = 2.5$ into equation (1): $a + 5(2.5) = 13$ $a + 12.5 = 13 \quad \implies \quad a = 0.5$

Final expression for $a_n$:

$a_n = 0.5 + (n-1)(2.5)$ Simplify: $a_n = 0.5 + 2.5n - 2.5$ $a_n = 2.5n - 2$

Final Answers:

(a) For the geometric sequence: $a_n = 3 \cdot 2^{n-1}$

(c) For the arithmetic sequence: $a_n = 2.5n - 2$

Would you like further clarification or a deeper explanation for any part? Here are 5 related questions to explore further:

1. How can we determine the nth term of a sequence from other given terms?
2. What is the difference between geometric and arithmetic sequences?
3. How can the common ratio of a geometric sequence be negative, and how does that affect the sequence?
4. What happens when you calculate a fractional or decimal $r$ in a geometric sequence?
5. Can we derive the sum of an arithmetic sequence given the formula for its nth term?

Tip: Always double-check calculations by substituting the values back into the sequence formulas to verify consistency!