"ONE!" a loud voice blares from your radio. "ONE!" it repeats. "TWO! THREE! FIVE!" the voice booms. Then it stops.
"It repeated '1' and it skipped '4,' just like the Fibonacci sequence (see "A Special Sequence")," says Anita.
"It sounds like the countdown that brought me into space (see "A Consequential Countdown")," you say. "Only this time the count went up instead of down."
"Hey, if the Fibonacci sequence brought you into space, maybe it also can bring us back to Earth somehow," says Bill.
"But the voice stopped counting," you say. "What if we are stuck here?"
Meanwhile, Anita is studying the numbers on the cubes that form Pascal's triangle (see "Pascal's Patterns"). "I bet the Fibonacci sequence is somewhere in Pascal's triangle," she says. "Every number pattern in the universe seems to show up in Pascal's triangle. If we could only find it, we might get somewhere."
The three of you start searching the numbered cubes for Fibonacci numbers, but they don't seem to line up together anywhere in Pascal's triangle. Can you find the Fibonacci sequence by adding together certain sets of numbers found in Pascal's triangle?
TRY IT!
Look for Fibonacci numbers in Pascal's triangle.
You will need:
- copy of Pascal's triangle (above)
- pencil
- ruler
What to do:
- On your diagram of Pascal's triangle, draw a series of diagonal lines, as shown below.
- For each diagonal line, add the numbers on the squares it passes through, and record the sum. Include only the squares it intersects more or less down the middle.
Looking for Fibonacci numbers in Pascal's triangle.
For example, the first line passes only through 1. So does the second line. The third line passes through two 1's, so it has a total of 2. The fourth line passes through 2 and 1, for a total of 3. Do you recognize the numbers?
Answers:
The fifth line adds up to 5. The next sum is 8, and the Fibonacci sequence continues with 13, 21, and so on.
Parting Ways
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