1. Яка з наведених послідовностей є арифметичною прогресією? А. $$ 16 ; 8 ; 4 ; 2 $$; $$ \qquad $$ Б. $$ -12 ;-6 ; 0 ; 6 ; $$ $$ \qquad $$ В. $$ 23 ; 51 ; 64 ; 82 $$; $$ \qquad $$ Г. $$ 3 ; 5 ; 7 ; 10 ; \ldots $$ 2. Яка з наведених послідовностей є геометричною прогресією? А. $$ 16 ;-8 ; 4 ;-2 $$; $$ \qquad $$ Б. $$ 2 ; 4 ; 12 ; 24 $$; $$ \qquad $$ В. $$ -1 ; 1 ; 0 ;-1 $$; $$ \qquad $$ Г. $$ 5 ; 8 ; 11 ; 14 ; \ldots $$ 3 . Знайдіть 18 -й член арифметичної прогресії $$ \left(a_{n}\right) $$, якщо $$ a_{3}=15, d=6 $$. А. 117. Б. 98. В. 105. Г. 111. 4. Знайдіть знаменник геометричної прогресії $$ \left(b_{n}\right) $$, якщо $$ b_{1}=2, b_{6}=486 $$. А. 4. Б. -3. В. 3. Г. 2. 5. Знайдіть суму перших дев'яти членів арифметичної прогресії $$ \left(a_{n}\right) $$, як. що $$ a_{1}=0,8, d=2 $$. А. 79,2 . Б. 72,9 . В. $$ 84,6 $$. Г. 75,96 . 6. Знайдіть суму перших п'яти членів геометричної прогресії $$ \left(b_{n}\right) $$, якщо $$ b_{1}=4, q=2 $$. А. 36. Б. 62. В. 128. Г. 124.

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Alright, I need to tackle these math problems related to arithmetic and geometric progressions. Let's go through each question one by one to ensure I understand and solve them correctly.

**Question 1:**
Which of the following sequences is an arithmetic progression?
A. 16 ; 8 ; 4 ; 2
B. -12 ; -6 ; 0 ; 6
C. 23 ; 51 ; 64 ; 82
D. 3 ; 5 ; 7 ; 10 ; …

First, let's recall that in an arithmetic progression, the difference between consecutive terms is constant. So, I'll calculate the differences for each sequence.

- **Sequence A:** 8 - 16 = -8, 4 - 8 = -4, 2 - 4 = -2. The differences are -8, -4, -2. These are not equal, so it's not an arithmetic progression.
- **Sequence B:** -6 - (-12) = 6, 0 - (-6) = 6, 6 - 0 = 6. The differences are all 6, so this is an arithmetic progression.
- **Sequence C:** 51 - 23 = 28, 64 - 51 = 13, 82 - 64 = 18. The differences are 28, 13, 18. These are not equal, so it's not an arithmetic progression.
- **Sequence D:** 5 - 3 = 2, 7 - 5 = 2, 10 - 7 = 3. The differences are 2, 2, 3. These are not equal, so it's not an arithmetic progression.

So, the correct answer is **B**.

**Question 2:**
Which of the following sequences is a geometric progression?
A. 16 ; -8 ; 4 ; -2
B. 2 ; 4 ; 12 ; 24
C. -1 ; 1 ; 0 ; -1
D. 5 ; 8 ; 11 ; 14 ; …

In a geometric progression, the ratio between consecutive terms is constant. Let's check each sequence.

- **Sequence A:** -8 / 16 = -0.5, 4 / -8 = -0.5, -2 / 4 = -0.5. The ratios are all -0.5, so this is a geometric progression.
- **Sequence B:** 4 / 2 = 2, 12 / 4 = 3, 24 / 12 = 2. The ratios are 2, 3, 2. These are not equal, so it's not a geometric progression.
- **Sequence C:** 1 / -1 = -1, 0 / 1 = 0, -1 / 0 is undefined. The ratios are -1, 0, undefined. This is not a geometric progression.
- **Sequence D:** 8 / 5 = 1.6, 11 / 8 = 1.375, 14 / 11 ≈ 1.272. The ratios are not equal, so it's not a geometric progression.

Therefore, the correct answer is **A**.

**Question 3:**
Find the 18th term of the arithmetic progression $$ (a_n) $$, given that $$ a_3 = 15 $$ and $$ d = 6 $$.
A. 117
B. 98
C. 105
D. 111

The formula for the nth term of an arithmetic progression is:
$$ a_n = a_1 + (n-1)d $$

First, I need to find $$ a_1 $$. I know $$ a_3 = 15 $$ and $$ d = 6 $$:
$$ a_3 = a_1 + 2d $$
$$ 15 = a_1 + 12 $$
$$ a_1 = 15 - 12 = 3 $$

Now, to find $$ a_{18} $$:
$$ a_{18} = a_1 + 17d $$
$$ a_{18} = 3 + 17 \times 6 $$
$$ a_{18} = 3 + 102 = 105 $$

So, the correct answer is **C**.

**Question 4:**
Find the common ratio of the geometric progression $$ (b_n) $$, given that $$ b_1 = 2 $$ and $$ b_6 = 486 $$.
A. 4
B. -3
C. 3
D. 2

The formula for the nth term of a geometric progression is:
$$ b_n = b_1 \times q^{(n-1)} $$

Given $$ b_6 = 486 $$ and $$ b_1 = 2 $$:
$$ 486 = 2 \times q^{5} $$
$$ q^{5} = 243 $$
$$ q = \sqrt[5]{243} $$
$$ q = 3 $$

So, the correct answer is **C**.

**Question 5:**
Find the sum of the first nine terms of the arithmetic progression $$ (a_n) $$, given that $$ a_1 = 0.8 $$ and $$ d = 2 $$.
A. 79.2
B. 72.9
C. 84.6
D. 75.96

The formula for the sum of the first n terms of an arithmetic progression is:
$$ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) $$

Plugging in the values:
$$ S_9 = \frac{9}{2} \times (2 \times 0.8 + 8 \times 2) $$
$$ S_9 = \frac{9}{2} \times (1.6 + 16) $$
$$ S_9 = \frac{9}{2} \times 17.6 $$
$$ S_9 = 4.5 \times 17.6 = 79.2 $$

So, the correct answer is **A**.

**Question 6:**
Find the sum of the first five terms of the geometric progression $$ (b_n) $$, given that $$ b_1 = 4 $$ and $$ q = 2 $$.
A. 36
B. 62
C. 128
D. 124

The formula for the sum of the first n terms of a geometric progression is:
$$ S_n = b_1 \times \frac{q^n - 1}{q - 1} $$

Plugging in the values:
$$ S_5 = 4 \times \frac{2^5 - 1}{2 - 1} $$
$$ S_5 = 4 \times \frac{32 - 1}{1} $$
$$ S_5 = 4 \times 31 = 124 $$

So, the correct answer is **D**.

1. B 2. A 3. C 4. C 5. A 6. D

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