Question

Сумма первых 14 членов арифметической прогрессии равна 427, а пятый член равен 23. Найдите отношение четырнадцатого члена к четвертому. Можно в подробностях

Answer 1

Решение во вложении:

Answer 2

Arithmetic Progression

To find the ratio between the fourteenth term and the fourth term of an arithmetic progression, we need to determine the common difference and the values of the first and fifth terms.

Given: - The sum of the first 14 terms of the arithmetic progression is 427. - The fifth term of the arithmetic progression is 23.

Let's break down the problem step by step.

Step 1: Finding the Common Difference

The sum of the first 14 terms of an arithmetic progression can be calculated using the formula:

Sum = (n/2)(2a + (n-1)d)

Where: - Sum is the sum of the terms, - n is the number of terms, - a is the first term, and - d is the common difference.

In this case, we know that the sum of the first 14 terms is 427. So we have:

427 = (14/2)(2a + (14-1)d)

Simplifying the equation:

427 = 7(2a + 13d) 61 = 2a + 13d

Step 2: Finding the First Term

To find the first term (a), we can use the formula:

a = t - (n-1)d

Where: - t is the term we know (in this case, the fifth term), - n is the position of the term, and - d is the common difference.

In this case, we know that the fifth term is 23. So we have:

a = 23 - (5-1)d a = 23 - 4d

Step 3: Solving the Equations

Now we have two equations:

Equation 1: 61 = 2a + 13d Equation 2: a = 23 - 4d

We can substitute Equation 2 into Equation 1 to solve for d:

61 = 2(23 - 4d) + 13d 61 = 46 - 8d + 13d 61 = 46 + 5d 5d = 61 - 46 5d = 15 d = 3

Substituting the value of d back into Equation 2, we can solve for a:

a = 23 - 4(3) a = 23 - 12 a = 11

Step 4: Finding the Fourteenth Term

Now that we know the values of a and d, we can find the fourteenth term using the formula:

t = a + (n-1)d

In this case, n = 14:

t = 11 + (14-1)(3) t = 11 + 13(3) t = 11 + 39 t = 50

Step 5: Finding the Ratio

Finally, we can find the ratio between the fourteenth term and the fourth term:

Ratio = Fourteenth Term / Fourth Term Ratio = 50 / (a + (4-1)d) Ratio = 50 / (11 + 3(4)) Ratio = 50 / (11 + 12) Ratio = 50 / 23

Therefore, the ratio between the fourteenth term and the fourth term is 50/23.

Please note that the calculations provided above are based on the given information and assumptions about the arithmetic progression.