Two starting numbers and one rule
The Fibonacci sequence is built by adding the previous two numbers to get the next one. Start with 0, 1, and the sequence continues 0, 1, 1, 2, 3, 5, 8, 13, 21.
Wolfram MathWorld gives the rule and starting values. You may also see a list begin 1, 1, 2, 3. That version leaves off the starting zero; it uses the same addition rule.
Build one number at a time
With 0 and 1 on the page, add them: 0 + 1 = 1. Now use the last two numbers, 1 and 1, to make 2.
Keep sliding along: 1 + 2 = 3, then 2 + 3 = 5, then 3 + 5 = 8. After 8 comes 5 + 8 = 13. Each new step needs the two most recent terms.
Try continuing after 13 and 21. You need 13 + 21 = 34. Adding 21 to itself would give 42, but that would be a different rule.
The jumps do not stay the same
A sequence such as 2, 4, 6, 8 adds two each time. Fibonacci does not. Between 3 and 5 the jump is two; between 5 and 8 it is three. What stays fixed is the instruction to add the previous two terms.
A few numbers can fit more than one possible rule. Here the starting values and the addition rule have been stated, so you can check every step instead of guessing from the first few terms. That builds on checking the whole number pattern.
Fibonacci numbers are often discussed alongside plant spirals, but a picture alone cannot prove a count. The sunflower cover is an illustration, not a measured example. For this lesson, the number-by-number calculation supplies the check.