Depth · one concept from first principles · step 2 of 4
Skip the lead-in, then add whole blocks
As a decimal, . Let be the digit in the th place after the decimal point of . Find the value of .
Number systemOperations
Your answer
How to solve
Strategy Look for a Pattern — Split the 45 places into the non-repeating start and the repeating part. The repeating part is a pattern of whole 6-digit blocks, so add one block and multiply.
1STEP 1
Separate the lead-in
Places 1, 2, 3 hold 0, 0, 0. That leaves places 4 through 45.
45 - 3 = 42
The zeros add nothing, and 42 digits remain.
8.NS.A.1Identify Subproblems2STEP 2
Count whole blocks
The repeating block 285714 has 6 digits, and 42 splits evenly into blocks of 6.
42 ÷ 6 = 7
Places 4 to 45 are exactly 7 copies of the block.
4.OA.C.5Look For A PatternPlaces through hold exactly whole copies of the block .
Why?
From place on, the -digit block repeats, so every places the digits start the block over again.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
The places split into groups of with nothing left over, because .
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
3STEP 3
Add one block, then multiply
One block's digits add to 27, and there are 7 blocks.
2 + 8 + 5 + 7 + 1 + 4 = 27, 27 × 7 = 189
The total digit sum is 189.
4.OA.C.5Look For A PatternAnswer
189
The average digit in a block is 27/6 = 4.5, and 42 digits at that average give 42 × 4.5 = 189, matching the block count.
Takeaway
Peel off the digits that don't repeat, then count in full blocks.
- Separate the lead-in
- Count whole blocks
- Add one block, then multiply
Where next?
Another one like thissuggested
Step back and solidifysuggested
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