Let's solve the given problems step by step.

(a) Arithmetic Progression

The general term for an arithmetic sequence is:

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

where:

• $a$ = first term
• $d$ = common difference
• $a_n$ = $n$-th term

Given:

• $a_8 = 57$
• $a_{17} = 111$

i. Find the Common Difference ($d$)

From the formula: $a_8 = a + 7d \quad \text{and} \quad a_{17} = a + 16d$

Subtract the two equations: $(a + 16d) - (a + 7d) = 111 - 57$ $9d = 54 \quad \Rightarrow \quad d = 6$

ii. Find the First Term ($a$)

Substitute $d = 6$ into $a_8 = a + 7d$: $57 = a + 7(6)$ $57 = a + 42 \quad \Rightarrow \quad a = 15$

iii. Sum of the First 28 Terms

The sum of the first $n$ terms of an arithmetic progression is:

$S_n = \frac{n}{2} \big(2a + (n-1)d\big)$

For $n = 28$: $S_{28} = \frac{28}{2} \big(2(15) + (28-1)(6)\big)$ $S_{28} = 14 \big(30 + 27(6)\big)$ $S_{28} = 14 \big(30 + 162\big) = 14(192) = 2688$

(b) Geometric Progression

The general term for a geometric sequence is:

$a_n = ar^{n-1}$

where:

• $a$ = first term
• $r$ = common ratio
• $a_n$ = $n$-th term

Given:

• $a_{10} = 177,147$
• $a_6 = 2,187$

i. Find the Common Ratio ($r$)

From the formula: $a_{10} = ar^9 \quad \text{and} \quad a_6 = ar^5$

Divide the two equations: $\frac{a_{10}}{a_6} = \frac{ar^9}{ar^5} \quad \Rightarrow \quad r^4 = \frac{177,147}{2,187}$

Simplify: $r^4 = 81 \quad \Rightarrow \quad r = \sqrt[4]{81} = 3$

ii. Find the First Term ($a$)

Substitute $r = 3$ into $a_6 = ar^5$: $2,187 = a(3^5)$ $2,187 = a(243) \quad \Rightarrow \quad a = \frac{2,187}{243} = 9$

Final Answers

(a) Arithmetic Progression

1. $d = 6$
2. $a = 15$
3. $S_{28} = 2688$

(b) Geometric Progression

1. $r = 3$
2. $a = 9$

Would you like a detailed explanation of any step or related examples?

Expanding Questions:

1. How would you calculate the $n$-th term of an arithmetic sequence with different values of $n$?
2. Can you derive the formula for the sum of a geometric series?
3. How would the solution change if the given terms in the sequences were decimals?
4. What is the relationship between $r$ and the growth rate in a geometric sequence?
5. How does changing $d$ or $r$ affect the behavior of the sequences?

Tip: When working with series, always ensure the values substituted for $n$, $a$, $d$, and $r$ match their respective equations to avoid errors.