How to write code to make the LED’s ﬂash in the following sequence?
int led_Pin_1 = 11;
int led_Pin_2 = 10;
int led_Pin_3 = 9;
int led_Pin_4 = 8;

int led_on = 0; // Start with first (led_Pin_1)

void setup() {
  pinMode(led_Pin_1, OUTPUT);
  pinMode(led_Pin_2, OUTPUT);
  pinMode(led_Pin_3, OUTPUT);
  pinMode(led_Pin_4, OUTPUT);
}

void loop() {
  // put your main code here, to run repeatedly:
  digitalWrite(led_Pin_1, (led_on == 0) ? HIGH : LOW);
  digitalWrite(led_Pin_2, (led_on == 1) ? HIGH : LOW);
  digitalWrite(led_Pin_3, (led_on == 2) ? HIGH : LOW);
  digitalWrite(led_Pin_4, (led_on == 3) ? HIGH : LOW);
  delay(250);

  led_on = (led_on + 1) % 4;
}


led_on will start with 0 (pin 1), switching only that pin on. Some explanation:

(led_on == 0) ? HIGH : LOW


This means: if (led_on == 0), than the value will be HIGH, otherwise LOW. Resulting in only pin 1 going on, if led_on is 0, and in the next loop, only pin 2 oing on, because led_on is 1 etc.

led_on = (led_on + 1) % 4;


This increases the value of led_on, but when it reaches 4, it reverts back to 0.

I don't know about the delay times you need. I guess you want to change all LEDs at once, than wait.

You can make the 4 almost identical digitalWrite lines into a function as small improvement.

If you want to do more with the LEDs, than consider storing the leds into an array and use a for loop to iterate through them.