Question

Числа u1 u2 u3 u4 сумма которыз равна 5 являются первыми четыремя членами геометрической прогрессии, а числа u2 u3 8/9u4 являются последовательными членами арифметической прогрессии найдите первый член и знаменатель геометрической прогрессии

Answer 1

u1 = p

u2 = pq

u3 = pq^2

u4 = pq^3

 

pq + 8/9 pq^3 = 2 pq^2 

8q^2 - 18q + 9 = 0

q_1 = 3/4, q_2 = 3/2

 

p(1 + q + q^2 + q^3) = 5

p = 5/ (1 + q + q^2 + q^3)

 

p_1 = 5/(1+3/4 + 9/16 + 27/64) = 5*64/(64+48+36+27) = 64/35

 

p_2 = 5/(1 + 3/2 + 9/4 + 27/8) = 5*8/(8+12+18+27) = 8/13

 

ответ

p1 = 64/35, q1 = 3/4

p2 = 8/13, q2 = 3/2

 

Answer 2

Problem Analysis

We are given that the numbers u1, u2, u3, and u4 form a geometric progression, and the numbers u2, u3, and 8/9u4 form an arithmetic progression. We need to find the first term (u1) and the common ratio (r) of the geometric progression.

Solution

Let's start by finding the common difference (d) of the arithmetic progression formed by u2, u3, and 8/9u4. We know that the common difference is the difference between consecutive terms in an arithmetic progression.

The common difference (d) can be found using the formula: d = u3 - u2

Next, we can find the value of u4 using the formula for the nth term of an arithmetic progression: u4 = u3 + d

Now, we have the values of u1, u2, u3, and u4. We can use these values to find the common ratio (r) of the geometric progression.

We know that the common ratio (r) can be found using the formula: r = u2 / u1

Finally, we can substitute the values of u2 and u1 into the equation r = u2 / u1 to find the common ratio (r). Once we have the common ratio, we can find the first term (u1) using the equation: u1 = u2 / r

Let's calculate the values step by step.

Calculation

Given: - u2, u3, u4 form a geometric progression - u2, u3, 8/9u4 form an arithmetic progression - The sum of u1, u2, u3, u4 is 5

Step 1: Find the common difference (d) of the arithmetic progression formed by u2, u3, and 8/9u4. d = u3 - u2

Step 2: Find the value of u4 using the formula for the nth term of an arithmetic progression. u4 = u3 + d

Step 3: Find the common ratio (r) of the geometric progression. r = u2 / u1

Step 4: Find the first term (u1) of the geometric progression. u1 = u2 / r

Let's calculate the values using the given information.

From the given information, we know that the sum of u1, u2, u3, and u4 is 5. Therefore, we can write the equation: u1 + u2 + u3 + u4 = 5

Substituting the values of u4 and d into the equation, we get: u1 + u2 + u3 + (u3 + d) = 5

Simplifying the equation, we have: 2u3 + 2d + u1 + u2 = 5

Since we have two equations involving u1, u2, u3, and u4, we can solve them simultaneously to find the values.

Let's solve the equations to find the values of u1, u2, u3, and u4.

Solution

From the given information, we have the following equations: 1. u1 + u2 + u3 + u4 = 5 2. d = u3 - u2 3. u4 = u3 + d 4. r = u2 / u1 5. u1 = u2 / r

We can solve these equations simultaneously to find the values of u1, u2, u3, and u4.

Using equation 2, we can express u3 in terms of u2 and d: u3 = u2 + d

Substituting this value of u3 into equation 3, we get: u4 = (u2 + d) + d = u2 + 2d

Now, we can substitute the expressions for u3 and u4 into equation 1: u1 + u2 + (u2 + d) + (u2 + 2d) = 5

Simplifying the equation, we have: 4u2 + 4d + u1 = 5

Next, we can substitute the expression for u1 from equation 5 into the above equation: (u2 / r) + 4u2 + 4d = 5

Multiplying both sides of the equation by r, we get: u2 + 4u2r + 4dr = 5r

Rearranging the terms, we have: 4u2r + (1 + 4d)r = 5r - u2

Factoring out r, we get: r(4u2 + 1 + 4d) = 5r - u2

Since r cannot be zero, we can divide both sides of the equation by r: 4u2 + 1 + 4d = 5 - (u2 / r)

Using equation 4, we can express u2 / r in terms of r: u2 / r = 1 / r

Substituting this value into the equation, we get: 4u2 + 1 + 4d = 5 - (1 / r)

Rearranging the terms, we have: 4u2 + 4d = 5 - 1 / r - 1

Simplifying the equation, we get: 4u2 + 4d = 4 - 1 / r

Now, we can substitute the expression for d from equation 2 into the above equation: 4u2 + 4(u3 - u2) = 4 - 1 / r

Simplifying the equation, we have: 4u3 - 4u2 = 4 - 1 / r

Using equation 3, we can express u3 in terms of u2 and d: u3 = u2 + d

Substituting this value into the equation, we get: 4(u2 + d) - 4u2 = 4 - 1 / r

Simplifying the equation, we have: 4u2 + 4d - 4u2 = 4 - 1 / r

The u2 terms cancel out, and we are left with: 4d = 4 - 1 / r

Now, we can substitute the expression for d from equation 2 into the above equation: 4(u3 - u2) = 4 - 1 / r

Simplifying the equation, we have: 4u3 - 4u2 = 4 - 1 / r

Using equation 1, we can express u4 in terms of u1, u2, and u3: u4 = 5 - u1 - u2 - u3

Substituting the expressions for u3 and u4 from equations 2 and 3 into the above equation, we get: u4 = 5 - u1 - u2 - (u2 + d)

Simplifying the equation, we have: u4 = 5 - u1 - 2u2 - d

Now, we can substitute the expression for d from equation 2 into the above equation: u4 = 5 - u1 - 2u2 - (u2 + (u3 - u2))

Simplifying the equation, we have: u4 = 5 - u1 - 2u2 - u3 + u2

The u2 terms cancel out, and we are left with: u4 = 5 - u1 - u3

Now, we have a system of equations that we can solve simultaneously to find the values of u1, u2, u3, and u4.

Let's solve the system of equations to find the values.

Solving the System of Equations

From the previous steps, we have the following system of equations: 1. 4u3 - 4u2 = 4 - 1 / r 2. u4 = 5 - u1 - u3

We can solve these equations simultaneously to find the values of u1, u2, u3, and u4.

Let's solve the system of equations.

From equation 1, we have: 4u3 - 4u2 = 4 - 1 / r

From equation 2, we have: u4 = 5 - u1 - u3

We can substitute the value of u4 from equation 2 into equation 1: 4u3 - 4u2 = 4 - 1 / r

Substituting the value of u4, we get: 4u3 - 4u2 = 4 - 1 / r = 5 - u1 - u3

Rearranging the terms, we have: 5u3 - 4u2 - u1 = 5 - 1 / r

Now, we can substitute the expression for r from equation 4 into the above equation: 5u3 - 4u2 - u1 = 5 - 1 / (u2 / u1)

Simplifying the equation, we have: 5u3 - 4u2 - u1 = 5 - u1 / u2

Multiplying both sides of the equation by u2, we get: 5u3u2 - 4u2^2 - u1u2 = 5u2 - u1

Rearranging the terms, we have: **5u3u2