Question

Задание ЕГЭ Математика -профиль №19 .У меня не сошлось с ответом ,поэтому прошу вас решить и скинуть более - менее подробное решение . Не к спеху "Все члены конечной последовательности являются натуральными числами. Каждый член этой последовательности, начиная со второго, либо в 8 раз больше, либо в 8 раз меньше предыдущего. Сумма всех членов последовательности равна 5984 . а)Может ли последовательность состоять из двух членов? б)Может ли последовательность состоять из трех членов? в)Какое наибольшее количество членов может быть последовательности ? "

Answer 1

А) Если последовательность состоит из двух членов a и 8a, то а+8а=5984. 
Уравнение 9а=5984 не имеет решения в натуральных числах. Поэтому последовательность не может состоять из 2-х членов. 
б) Нет, не может. Т.к нет таких чисел, которые дадут при сложении 4 на конце. 
в) Приведем пример последовательности из 1330 членов. 8,1,8,1.....
Сумма ее членов равна 8+9*664 = 5984

Answer 2

Task Analysis

To solve this problem, we need to determine the maximum number of terms in a sequence that satisfies the given conditions. The sequence is defined as follows: - All terms are natural numbers. - Starting from the second term, each term is either 8 times greater or 8 times smaller than the previous term. - The sum of all terms in the sequence is 5984.

We need to answer the following questions: a) Can the sequence consist of only two terms? b) Can the sequence consist of only three terms? c) What is the maximum number of terms that the sequence can have?

Solution

a) To determine if the sequence can consist of only two terms, we need to check if there exists a pair of numbers whose sum is 5984 and one is 8 times greater or smaller than the other. Let's assume the two terms are x and y, where y is 8 times greater or smaller than x.

If y is 8 times greater than x, we have the equation: x + 8x = 5984 Simplifying the equation, we get: 9x = 5984 x = 5984 / 9 x ≈ 665.33

Since x is a natural number, it cannot be a decimal. Therefore, the sequence cannot consist of only two terms.

b) To determine if the sequence can consist of only three terms, we need to check if there exists a triplet of numbers whose sum is 5984 and each term is 8 times greater or smaller than the previous term. Let's assume the three terms are x, y, and z, where y is 8 times greater or smaller than x, and z is 8 times greater or smaller than y.

If y is 8 times greater than x and z is 8 times greater than y, we have the equations: x + 8x + 8(8x) = 5984 17x = 5984 x = 5984 / 17 x ≈ 352

Since x is a natural number, it is possible for the sequence to consist of three terms.

c) To find the maximum number of terms in the sequence, we can start with the assumption that the first term is 1. Then, we can iteratively calculate the sum of the sequence by multiplying each term by 8 and adding it to the sum. We continue this process until the sum exceeds 5984. The number of terms in the sequence will be one less than the number of iterations performed.

Let's calculate the maximum number of terms in the sequence:

1. First term: 1 2. Sum: 1 3. Iteration 1: 1 * 8 = 8, sum = 1 + 8 = 9 4. Iteration 2: 8 * 8 = 64, sum = 9 + 64 = 73 5. Iteration 3: 64 * 8 = 512, sum = 73 + 512 = 585 6. Iteration 4: 512 * 8 = 4096, sum = 585 + 4096 = 4681 7. Iteration 5: 4096 * 8 = 32768, sum = 4681 + 32768 = 37449

The sum has exceeded 5984 after 5 iterations. Therefore, the maximum number of terms in the sequence is 5.

Answer

a) The sequence cannot consist of only two terms. b) The sequence can consist of only three terms. c) The maximum number of terms in the sequence is 5.

Please note that the calculations provided above are based on the given conditions and assumptions.