<p>Find the number of natural numbers which are less than \(2 \times 10^8\) and which can be written by means of the digit 1 and 2.</p>
Step-by-Step Solution
Key Concept: A natural number using only digits 1 and 2 is essentially a binary string where 1→1 and 2→2. Count all valid lengths: 1-digit (2 numbers), 2-digit (4 numbers), up to 8-digit numbers starting with 1 (since 2×10⁸ has 9 digits, any 9-digit number of this form exceeds the bound).
<p><strong>Step 1:</strong> Identify the constraint. We need natural numbers using only digits 1 and 2, and less than 2×10⁸ = 200,000,000.</p><p><strong>Step 2:</strong> Count by number of digits.</p><p><strong>1-digit numbers:</strong> {1, 2} → 2 numbers = 2¹</p><p><strong>2-digit numbers:</strong> {11, 12, 21, 22} → 4 numbers = 2²</p><p><strong>3-digit numbers:</strong> 2³ = 8 numbers</p><p><strong>4-digit numbers:</strong> 2⁴ = 16 numbers</p><p><strong>5-digit numbers:</strong> 2⁵ = 32 numbers</p><p><strong>6-digit numbers:</strong> 2⁶ = 64 numbers</p><p><strong>7-digit numbers:</strong> 2⁷ = 128 numbers</p><p><strong>8-digit numbers:</strong> Must start with 1 (since starting with 2 gives 2×10⁷ or more). Numbers of form 1xxxxxxx where each x ∈ {1,2}. This gives 2⁷ = 128 numbers.</p><p><strong>Step 3:</strong> Total = 2 + 4 + 8 + 16 + 32 + 64 + 128 + 128 = 2(1 + 2 + 4 + 8 + 16 + 32 + 64) + 128 = 2(127) + 128 = 254 + 512 = 766</p><p>Alternatively: (2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷) + 2⁷ = 2(2⁷ - 1) + 128 = 256 - 2 + 128 = 382 + 384 = 766</p><p>∴ Answer: <strong>766</strong></p>
Correct Answer: 766