Question

Пожалуйста помогите! Задачка на Английском языке! An infinite geometric series has a common ratio, r. The first term of the series is 10 and the sum to infinity is 50. (a) Show that the common ratio is 4/5 (b) Find the second term of the series (c) The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms respectively, in an arithmetic series. Find the common difference of the arithmetic series.

Answer 1

b₁=10 S=50 q<|1|

a) доказать, что q=4/5.

S=b₁/(1-q)=50

10/(1-q)=50

50*(1-q)=10

50-50q=10

50q=40 |÷50

q=4/5.

b) найти второй член прогрессии:

b₂=b₁*q=10*4/5=8.

b₂=8.

с)

a₄=10     a₁+3d=10

a₈=8       a₁+7d=8

Вычитаем из второго уравнения первое:

4d=-2   |÷4

d=-0,5.

Answer 2

Решение во вложении.

Answer 3

Sure, I'd be happy to help you with this problem!

Let's go step by step:

(a) Show that the common ratio is 4/5: In an infinite geometric series, if the first term is "a" and the common ratio is "r," then the sum to infinity is given by the formula: Sum = a / (1 - r)

Given that the first term (a) is 10 and the sum to infinity is 50, we can write the equation as follows: 50 = 10 / (1 - r)

Now, we need to solve for "r": Multiply both sides by (1 - r): 50 * (1 - r) = 10

Expand the left side: 50 - 50r = 10

Move -50r to the right side: 50 - 10 = 50r

Simplify: 40 = 50r

Now, divide both sides by 50: r = 40 / 50

Reduce the fraction: r = 4/5

So, we have shown that the common ratio (r) is 4/5.

(b) Find the second term of the series: In a geometric series, if the first term is "a" and the common ratio is "r," then the second term (let's call it "b") can be found using the formula: b = a * r

Given that the first term (a) is 10 and the common ratio (r) is 4/5, we can find the second term as follows: b = 10 * (4/5) b = 8

The second term of the series is 8.

(c) The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms, respectively, in an arithmetic series. Find the common difference of the arithmetic series.

In an arithmetic series, if the first term is "a" and the common difference is "d," then the nth term (Tn) can be found using the formula: Tn = a + (n - 1) * d

We know that the first term (a) of the arithmetic series is the same as the first term (10) of the geometric series, and the second term (b) of the geometric series is the same as the fourth term of the arithmetic series. So, we can set up the following equations:

a (first term of arithmetic series) = 10 b (second term of arithmetic series) = 8 n (corresponding term in arithmetic series) = 4

Using the formula for the nth term of an arithmetic series, we have: b = a + (n - 1) * d 8 = 10 + (4 - 1) * d 8 = 10 + 3d

Now, solve for "d": 8 - 10 = 3d -2 = 3d

Finally, divide by 3: d = -2 / 3

So, the common difference of the arithmetic series is -2/3.