<p>The sequence of all positive palindromes are written in ascending order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, ...</p><p>Then the <math>2018</math>th positive palindrome is</p>
Step-by-Step Solution
Key Concept: Count palindromes systematically by number of digits: 1-digit has 9, 2-digit has 9, 3-digit has 90, etc. Find which digit-length contains the 2018th palindrome, then determine its exact value within that group.
<p><strong>Step 1: Count palindromes by digit length</strong></p><p>• 1-digit palindromes: 1,2,...,9 → 9 palindromes</p><p>• 2-digit palindromes: 11,22,...,99 → 9 palindromes</p><p>• 3-digit palindromes: 101,111,...,191,202,... → First digit (1-9) and middle digit (0-9) determine it → 9×10 = 90 palindromes</p><p>• 4-digit palindromes: First two digits (10-99) determine it → 90 palindromes</p><p>• 5-digit palindromes: First digit (1-9) and next two digits (00-99) → 9×100 = 900 palindromes</p><p>• 6-digit palindromes: First three digits (100-999) → 900 palindromes</p><p><strong>Step 2: Find cumulative count</strong></p><p>Total through 6-digit palindromes: 9 + 9 + 90 + 90 + 900 + 900 = 1998</p><p><strong>Step 3: Determine position in 7-digit palindromes</strong></p><p>We need the 2018th palindrome. Since we have 1998 through 6-digit, we need the (2018 - 1998) = 20th palindrome among 7-digit palindromes.</p><p><strong>Step 4: Structure of 7-digit palindromes</strong></p><p>A 7-digit palindrome has form: abcdcba, where the first 4 digits (abcd) determine it completely.</p><p>The first digit 'a' ranges from 1-9, and the next three digits 'bcd' range from 000-999.</p><p>These are ordered as: 1000001, 1001001, 1002001, ..., 1009001, 1010101, ...</p><p><strong>Step 5: Find the 20th seven-digit palindrome</strong></p><p>• 1st to 10th: a=1, bcd = 000 to 009 → forms 1000001, 1001001, ..., 1009001</p><p>• 11th to 20th: a=1, bcd = 010 to 019 → forms 1010101, 1011101, ..., 1019101</p><p>The 20th palindrome corresponds to a=1, bcd=019, giving 1019101.</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A