Question

Помогите понять арифметическую последовательность. Например, задана следующая: 3+5+7+...+401 Необходимо найти сумму всех членов. Формула известна, однако не ясно, почему это работает. находим шаг прогрессии, окей, легко +2. Затем пытаемся найти количество вмещений этой двойки в последний член, учитывая, что первый = 3. А вот сейчас совсем не ясно, исходя из формулы (An-A1) / 2 = 199 и мы прибавляем 1, и того, = 200, что верный ответ и позволяет найти сумму, однако вопрос, как мы можем изъять из прогрессии тройку, чтобы добавить её вроде бы, как дополнительный шаг прогрессии (Наше +1), если шаг прогрессии = 2. Выглядит так, будто единичку мы потеряли, на деле же - нет. Почему?

Answer 1

Формула такова
$sn = \frac{a1 + an}{2} \times n$
получается так
$\frac{3 + 401}{2} \times 401 = 81002$
Формула, как найти сумму членов арифметической прогрессии.

Шаг прогрессии, как вы сказали, нужен, чтобы найти неизвестный член прогрессии, а в сумме он не нужен.

Answer 2

Understanding an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. To find the sum of all the terms in an arithmetic sequence, you can use the formula:

Sum = (n/2)(a1 + an)

Where: - Sum is the sum of all the terms. - n is the number of terms in the sequence. - a1 is the first term. - an is the last term.

In the given example, the arithmetic sequence is: 3, 5, 7, ..., 401. The first term, a1, is 3, and the last term, an, is 401. We need to find the sum of all the terms.

Finding the Number of Terms

To find the number of terms, n, in the sequence, we can use the formula:

n = (an - a1) / d + 1

Where: - d is the common difference between consecutive terms.

In this case, the common difference, d, is 2. Substituting the values into the formula, we get:

n = (401 - 3) / 2 + 1 = 200

So, there are 200 terms in the sequence.

Calculating the Sum

Now that we know the number of terms, we can calculate the sum using the formula mentioned earlier:

Sum = (n/2)(a1 + an)

Substituting the values, we get:

Sum = (200/2)(3 + 401) = 100(404) = 40,400

Therefore, the sum of all the terms in the arithmetic sequence 3, 5, 7, ..., 401 is 40,400.

Explanation of the Missing Term

You mentioned that it seems like we are missing the term 1 in the sequence, given that the common difference is 2. However, in an arithmetic sequence, the common difference refers to the difference between consecutive terms, not the actual terms themselves.

In this case, the common difference is 2, so each term is obtained by adding 2 to the previous term. The sequence starts with 3, and each subsequent term is obtained by adding 2. Therefore, the sequence does not include the number 1.

I hope this explanation helps clarify your understanding of arithmetic sequences. Let me know if you have any further questions!