Question

Сумма первого и третьего членов геометрической прогрессии равна 15, а сумма второго и четвертого равна30. Найти знаменатель геометрической прогрессии.​

Answer 1

Ответ:

b1+b3=15

b2+b4=30

S10=?

                b1*(1 - q^n)

S10 =   ---------------------

                   q - n

b1+b3=15

b2+b4=30

b2=b1*q

b3=b1*q²

b4=b1*q³

b1+b1*q² =15

b1*q + b1*q³=30

b1(1+q²)=15

b1(q+q³)=30

Поделим 2 на 1

b1(q+q³)       30

------------ = ------

b1(1+q²)      15

q + q³                                           q (1 + q²)

---------- = 2                                  -------------- = 2  

1 + q²                                               1 + q²  

q = 2

   

b1(1+q²) =15          

b1(1+ 2²)=15

5b1 = 15

b1 = 3

             

                3 * (1 - 2^10)            3 * (1 - 1024)

S10 =  ---------------------- =    ---------------------

                   1 - 2                               -1

S10 = 3 * 1023 = 3069

S10 = 3069

Пошаговое объяснение:

Answer 2

Problem Analysis

We are given a geometric progression and we need to find the common ratio of the progression. We are given two sums: the sum of the first and third terms is 15, and the sum of the second and fourth terms is 30.

Solution

Let's assume that the first term of the geometric progression is a and the common ratio is r.

The sum of the first and third terms can be expressed as: a + ar^2 = 15 The sum of the second and fourth terms can be expressed as: ar + ar^3 = 30 To solve this system of equations, we can use substitution or elimination.

Substitution Method

We can solve the first equation for a and substitute it into the second equation:

a = 15 - ar^2

Substituting this into the second equation:

(15 - ar^2)r + ar^3 = 30

Simplifying:

15r - ar^3 + ar^3 = 30

15r = 30

r = 2

Therefore, the common ratio of the geometric progression is 2.

Verification

To verify our solution, let's substitute the common ratio r = 2 into the original equations:

a + a(2)^2 = 15

a + 4a = 15

5a = 15

a = 3

The first term of the geometric progression is 3.

Now, let's check the second equation:

a(2) + a(2)^3 = 30

6 + 24 = 30

The sum of the second and fourth terms is indeed 30.

Therefore, our solution is verified.

Answer

The common ratio of the geometric progression is 2.