Question

Леша не поленился вычислить сумму 9+99+999+...+9...9(2017) и выписать ее на доску .Сколько раз в итоговом результате записана цифра 1? ПОМОГИТЕ ПОЖАЛУЙСТА!!!!!!!!!!! СРОЧНО!!!!!!

Answer 1

Ответ^^^^^^_^^^^^^^^_^^^^^^^_^^^^^^^

Answer 2

Calculation of the Sum

To calculate the sum of the series 9 + 99 + 999 + ... + 9...9 (2017 times), we can observe that each term in the series is formed by concatenating the digit 9 a certain number of times. The number of times the digit 9 is repeated in each term follows a pattern: the first term has 1 digit 9, the second term has 2 digits 9, the third term has 3 digits 9, and so on.

Let's denote the number of digits 9 in each term as n. We can see that the nth term can be expressed as 9 * (10^n - 1) / 9, which simplifies to 10^n - 1. Therefore, the sum of the series can be calculated as the sum of all these terms:

9 + 99 + 999 + ... + 9...9 (2017 times) = (10^1 - 1) + (10^2 - 1) + (10^3 - 1) + ... + (10^2017 - 1)

Counting the Number of 1s

To count the number of times the digit 1 appears in the final result, we need to analyze the individual terms in the series. Let's consider the general term (10^n - 1).

When n = 1, the term is (10^1 - 1) = 9. There are no occurrences of the digit 1 in this term.

When n = 2, the term is (10^2 - 1) = 99. There are two occurrences of the digit 1 in this term.

When n = 3, the term is (10^3 - 1) = 999. There are three occurrences of the digit 1 in this term.

We can observe that for each term, the number of occurrences of the digit 1 is equal to the value of n.

Therefore, in the final result, the number of times the digit 1 appears is equal to the sum of the first 2017 positive integers:

1 + 2 + 3 + ... + 2017 = (2017 * (2017 + 1)) / 2 = 2037178.

So, the digit 1 appears 2,037,178 times in the final result.

Note: The above calculation assumes that the series starts with the term 9. If the series starts with a different number, the calculation would be different.