Question

Сумма первых десяти членов АРИФМЕТИЧЕСКОЙ прогрессии равна 30. ЧЕТВЁРТЫЙ,СЕДЬМОЙ,ПЯТЫЙ её члены в УКАЗАННОМ порядке составляют ГЕОМЕТРИЧЕСКУЮ прогрессию. Найти разность арифметической прогрессии,если известно,что все её члены РАЗЛИЧНЫ

Answer 1

10a1+d*45=30

a1+3d

a1+6d

a1+4d

(a1+3d)(a1+4d)=(a1+6d)^

a1^+12d^+a1d*7=a1^+36d^+12a1d

5a1d=-24d^

5a1=-24d

a1=-4,8d

10a1+d*45=30

-48d+45d=-3d=30

d=-10

 

Answer 2

Arithmetic Progression (AP) and Geometric Progression (GP)

To find the difference of an arithmetic progression (AP) when the fourth, seventh, and fifth terms form a geometric progression (GP), we need to use the given information and solve for the values.

Let's break down the problem step by step:

1. We are given that the sum of the first ten terms of the arithmetic progression is 30. This means that the sum of the terms from the first term to the tenth term is 30.

2. We are also given that the fourth, seventh, and fifth terms of the arithmetic progression form a geometric progression. This means that the ratio between the seventh term and the fourth term is the same as the ratio between the fifth term and the fourth term.

Now, let's solve for the values:

1. Sum of the first ten terms of the arithmetic progression: - The sum of an arithmetic progression can be calculated using the formula: \(S_n = \frac{n}{2}(2a + (n-1)d)\), where \(S_n\) is the sum of the first \(n\) terms, \(a\) is the first term, and \(d\) is the common difference. - In this case, we are given that the sum is 30 and the number of terms is 10. So we have the equation: \(30 = \frac{10}{2}(2a + (10-1)d)\). - Simplifying the equation, we get: \(30 = 5(2a + 9d)\). - This equation can be further simplified to: \(6a + 27d = 6\).

2. Geometric progression formed by the fourth, seventh, and fifth terms: - Let's assume that the fourth term is \(a\), the common ratio is \(r\), and the seventh term is \(ar^3\) (since the seventh term is the fourth term multiplied by \(r\) three times). - We are given that the ratio between the seventh term and the fourth term is the same as the ratio between the fifth term and the fourth term. So we have the equation: \(\frac{ar^3}{a} = \frac{ar^2}{a}\). - Simplifying the equation, we get: \(r^3 = r^2\). - Since all the terms of the arithmetic progression are distinct, we know that \(r\) cannot be equal to 1. Therefore, we can divide both sides of the equation by \(r^2\) to get: \(r = 1\).

Now that we have the value of \(r\), we can solve for the common difference (\(d\)) of the arithmetic progression:

1. Substitute \(r = 1\) into the equation \(6a + 27d = 6\): - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\) - \(6a + 27d = 6\)