Step by step
The question
Question: The pattern up, up, down, up, up, down, ...
keeps repeating.
What is the 40th step?
1 Find the block
The list starts again after down.
So the block that repeats is up, up, down.
block = up, up, down
The block is 3 long.
length = 3
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
n ÷ (length of the block): what is left = the place in the block
3 times 13 is 39.
So 13 full blocks take us to the 39th step.
40 minus 39 is 1 more.
40 ÷ 3 = 3 × 13 = 39, 40 − 39 = 1
3 Count into the block
Careful!
The 13 only counts full blocks.
The 1 left over gives the place.
Careful: not 13: the number of blocks ✗
1 left means place 1 in the block: up.
40th step = place 1 in the block = up
4 Check
Let's check by counting on.
A block ends at the 39th step.
Then we count on: up.
The 40th is up.
So it is right!
39th = down (end of a block), 40th = up ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer
40th step = place 1 in the block = up