Question

Является ли последовательность 1/4, 1/2, 1, 2, ... геометрической прогрессией? Если да, то найдите её 8-й член; 10-й член; сумму первых восьми членов.

Answer 1

Ответ:

$\dfrac{1}{4}\ ,\ \ \dfrac{1}{2}\ ,\ \ 1\ ,\ \ 2\ ,\ ...\\\\\dfrac{b_2}{b_1}=\dfrac{1/2}{1/4}=2\ \ ,\ \ \ \dfrac{b_3}{b_2}=\dfrac{1}{1/2}=2\ \ ,\ \ \ \dfrac{b_4}{b_3}=\dfrac{2}{1}=2\ \ \ \Rightarrow \ \ \ q=2$

да, эта последовательность - геометрическая прогрессия

$b_8=b_1q^7=\dfrac{1}{4}\cdot 2^7=2^5=32\\\\b_{10}=b_8\cdot q^2=32\cdot 2^2=128\ \ \ (ili\ \ b_{10}=b_1q^9=\dfrac{1}{4}\cdot 2^9=128)\\\\S_8=\dfrac{b_8q-b_1}{q-1}=\dfrac{32\cdot 2-\frac{1}{4}}{2-1}=\dfrac{64\cdot 4-1}{4}=\dfrac{255}{4}=63\dfrac{3}{4}=63,75$

Answer 2

Is the sequence 1/4, 1/2, 1, 2, ... a geometric progression?

Yes, the sequence 1/4, 1/2, 1, 2, ... is a geometric progression.

A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio is 2, as each term is obtained by multiplying the previous term by 2.

Finding the 8th term of the geometric progression:

To find the 8th term of the geometric progression, we can use the formula for the nth term of a geometric progression:

nth term = a * r^(n-1)

where: - a is the first term of the sequence - r is the common ratio - n is the position of the term we want to find

In this case, the first term (a) is 1/4, the common ratio (r) is 2, and we want to find the 8th term (n = 8).

Plugging these values into the formula, we get:

8th term = (1/4) * 2^(8-1)

Simplifying the expression, we have:

8th term = (1/4) * 2^7 = (1/4) * 128 = 32

Therefore, the 8th term of the geometric progression is 32.

Finding the 10th term of the geometric progression:

Using the same formula as above, we can find the 10th term of the geometric progression.

10th term = (1/4) * 2^(10-1)

Simplifying the expression, we have:

10th term = (1/4) * 2^9 = (1/4) * 512 = 128

Therefore, the 10th term of the geometric progression is 128.

Finding the sum of the first eight terms of the geometric progression:

To find the sum of the first eight terms of a geometric progression, we can use the formula for the sum of a geometric series:

sum = a * (1 - r^n) / (1 - r)

where: - a is the first term of the sequence - r is the common ratio - n is the number of terms we want to sum

In this case, the first term (a) is 1/4, the common ratio (r) is 2, and we want to sum the first eight terms (n = 8).

Plugging these values into the formula, we get:

sum = (1/4) * (1 - 2^8) / (1 - 2)

Simplifying the expression, we have:

sum = (1/4) * (1 - 256) / (-1) = (1/4) * (-255) = -255/4

Therefore, the sum of the first eight terms of the geometric progression is -255/4.