Step by step
The question
Question: The pattern 2, 5, 9, 2, 5, 9, ...
keeps repeating.
What is the 100th number?
1 Find the block
The list starts again after 9.
So the block that repeats is 2, 5, 9.
block = 2, 5, 9
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 33 is 99.
So 33 full blocks take us to the 99th number.
100 minus 99 is 1 more.
100 ÷ 3 = 3 × 33 = 99, 100 − 99 = 1
3 Count into the block
Careful!
The 33 only counts full blocks.
The 1 left over gives the place.
Careful: not 33: the number of blocks ✗
1 left means place 1 in the block: 2.
100th number = place 1 in the block = 2
4 Check
Let's check by counting on.
A block ends at the 99th number.
Then we count on: 2.
The 100th is 2.
So it is right!
99th = 9 (end of a block), 100th = 2 ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer
100th number = place 1 in the block = 2