Let’s solve this step by step.

(a) Arithmetic Progression (AP)

Given:

• Second term: $a_2 = 7$
• Thirteenth term: $a_{13} = 10 \times a_1$

The formula for the $n$-th term of an arithmetic progression is:
$a_n = a_1 + (n - 1) \cdot d$
where $a_1$ is the first term and $d$ is the common difference.

• For the second term:
  $a_2 = a_1 + (2 - 1) \cdot d = a_1 + d = 7$ Thus: $a_1 + d = 7 \quad \text{(Equation 1)}$

• For the thirteenth term:
  $a_{13} = a_1 + (13 - 1) \cdot d = a_1 + 12d$ It is given that: $a_{13} = 10 \times a_1$ Thus: $a_1 + 12d = 10a_1 \quad \text{(Equation 2)}$ Rearrange Equation 2: $12d = 10a_1 - a_1 = 9a_1 \quad \Rightarrow \quad d = \frac{9a_1}{12} = \frac{3a_1}{4}$

Now solve for $a_1$ and $d$:

Substitute $d = \frac{3a_1}{4}$ into Equation 1: $a_1 + \frac{3a_1}{4} = 7$ $\frac{4a_1 + 3a_1}{4} = 7 \quad \Rightarrow \quad \frac{7a_1}{4} = 7$ $7a_1 = 28 \quad \Rightarrow \quad a_1 = 4$

Now, substitute $a_1 = 4$ into Equation 1: $4 + d = 7 \quad \Rightarrow \quad d = 3$

Thus, the first term is $a_1 = 4$ and the common difference is $d = 3$.

(b) Geometric Series (GS)

Given:

• Second term: $a_2 = 5$
• Sum to infinity: $S_\infty = 20$

The formula for the sum to infinity of a geometric series is: $S_\infty = \frac{a_1}{1 - r}$ where $a_1$ is the first term and $r$ is the common ratio ($|r| < 1$).

Also, the formula for the second term is: $a_2 = a_1 \cdot r$ Thus: $5 = a_1 \cdot r \quad \Rightarrow \quad a_1 = \frac{5}{r} \quad \text{(Equation 3)}$

Now, substitute $a_1 = \frac{5}{r}$ into the formula for $S_\infty$: $20 = \frac{\frac{5}{r}}{1 - r}$ Simplify: $20 = \frac{5}{r(1 - r)}$ $5 = 20r(1 - r)$ $5 = 20r - 20r^2 \quad \Rightarrow \quad 20r^2 - 20r + 5 = 0$ Divide by 5: $4r^2 - 4r + 1 = 0$ This is a quadratic equation in $r$. Solve using the quadratic formula: $r = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)} = \frac{4 \pm \sqrt{16 - 16}}{8} = \frac{4 \pm 0}{8} = \frac{4}{8} = \frac{1}{2}$ Thus, $r = \frac{1}{2}$.

Now, substitute $r = \frac{1}{2}$ into Equation 3: $a_1 = \frac{5}{\frac{1}{2}} = 5 \times 2 = 10$

Thus, the first term is $a_1 = 10$ and the common ratio is $r = \frac{1}{2}$.

Final Answers:

(a)

• First term: $a_1 = 4$
• Common difference: $d = 3$

(b)

• First term: $a_1 = 10$
• Common ratio: $r = \frac{1}{2}$

Do you want further details or have any questions?
Here are five follow-up questions to expand your understanding:

1. How do we derive the sum formula for an arithmetic progression?
2. What happens if the common ratio in a geometric series is greater than 1?
3. How do we determine the number of terms in a finite geometric series?
4. Can a geometric series with a negative common ratio have a sum to infinity?
5. What are the conditions for an arithmetic sequence to have a negative common difference?

Tip:
When solving quadratic equations, always check if the discriminant is non-negative to ensure real solutions exist.