Let's analyze and solve the given problems based on the image provided.

Problem (a)

Hasil tambah $n$ sebutan pertama bagi suatu janjang aritmetik diberikan oleh $S_n = 3n^2 + 5n$. Cari (i) Sebutan pertama dan beza sepunya. (ii) Hasil tambah dari sebutan ke-10 hingga ke sebutan ke-20.

(i) Sebutan pertama dan beza sepunya:

The sum of the first $n$ terms $S_n$ of an arithmetic progression (AP) is given by: $S_n = 3n^2 + 5n$

To find the first term $a$ and the common difference $d$, we use: $S_1 = a$ $S_2 = a + (a + d) = 2a + d$

From $S_n = 3n^2 + 5n$, we get: $S_1 = 3(1)^2 + 5(1) = 3 + 5 = 8$ So, $a = 8$.

For $S_2$: $S_2 = 3(2)^2 + 5(2) = 3 \cdot 4 + 10 = 12 + 10 = 22$

The second term can also be written as: $S_2 = a + (a + d) = 2a + d$ $22 = 2(8) + d$ $22 = 16 + d$ $d = 6$

Therefore, the first term $a$ is 8, and the common difference $d$ is 6.

(ii) Hasil tambah dari sebutan ke-10 hingga ke sebutan ke-20:

We need to find the sum from the 10th term to the 20th term. This can be written as: $S_{10 \text{ to } 20} = S_{20} - S_{9}$

Using $S_n = 3n^2 + 5n$: $S_{20} = 3(20)^2 + 5(20) = 3(400) + 100 = 1200 + 100 = 1300$ $S_9 = 3(9)^2 + 5(9) = 3(81) + 45 = 243 + 45 = 288$

Thus, the sum from the 10th to the 20th term is: $S_{10 \text{ to } 20} = 1300 - 288 = 1012$

Problem (b)

Tiga sebutan pertama bagi suatu janjang geometri ialah $x + 1, 3x, 5x + 2$. Cari nilai $x$.

The first three terms of a geometric progression (GP) are given as $x + 1, 3x, 5x + 2$.

For a GP, the ratio between successive terms is constant. Thus: $\frac{3x}{x + 1} = \frac{5x + 2}{3x}$

Cross-multiplying gives: $3x(3x) = (x + 1)(5x + 2)$ $9x^2 = 5x^2 + 2x + 5x + 2$ $9x^2 = 5x^2 + 7x + 2$ $4x^2 - 7x - 2 = 0$

Solving this quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $a = 4, \, b = -7, \, c = -2$ $x = \frac{7 \pm \sqrt{49 + 32}}{8}$ $x = \frac{7 \pm \sqrt{81}}{8}$ $x = \frac{7 \pm 9}{8}$

Thus, $x$ can be: $x = \frac{16}{8} = 2 \quad \text{or} \quad x = \frac{-2}{8} = -\frac{1}{4}$

Problem (c)

Hitung hasil tambah bagi 10 sebutan pertama.

This problem appears to be a continuation, but let's calculate the sum of the first 10 terms of the arithmetic progression given in (a).

The sum of the first 10 terms $S_{10}$ for the AP is: $S_{10} = 3(10)^2 + 5(10) = 3(100) + 50 = 300 + 50 = 350$

Would you like more details or have any questions?

Here are five related questions to further your understanding:

1. What is the sum of the first 15 terms of the arithmetic progression?
2. How do you derive the general term of an arithmetic progression from its sum formula?
3. What are the steps to solve a quadratic equation like $4x^2 - 7x - 2 = 0$ using the quadratic formula?
4. How do you find the sum of the first $n$ terms of a geometric progression?
5. What are the properties of a geometric progression that differentiate it from an arithmetic progression?

Tip: When dealing with sequences and series, always ensure you understand the formulas for the sum of terms and the general term. These are crucial for solving most problems involving progressions.