<p>In the 20th row of the triangle<br>\[\begin{array}{ccccc} & & 1 & & \\ & 2 & & 3 & \\ 4 & & 5 & & 6 \\ 7 & & 8 & & 9 & & 10 \\ \vdots & & & & & & \ddots \end{array}\]</p>
Step-by-Step Solution
Key Concept: The nth row contains n numbers. Find which number starts the 20th row by summing 1+2+3+...+19 (total numbers in first 19 rows), then identify the specific position within that row using the arithmetic progression pattern.
<p><strong>Step 1: Find total numbers in first 19 rows</strong></p><p>Row n contains n numbers. Total numbers in first 19 rows = 1 + 2 + 3 + ... + 19 = 19(20)/2 = 190</p><p><strong>Step 2: Find the starting number of row 20</strong></p><p>The 20th row starts at the 191st number overall, so it begins with 191</p><p><strong>Step 3: Identify the pattern in row 20</strong></p><p>Row 20 contains 20 numbers. Based on the triangle pattern, row n has numbers at positions that follow the sequence. The 20th row contains: 191, 192, 193, ..., 210</p><p><strong>Step 4: Find the first (or key) number in row 20</strong></p><p>The first number in the 20th row is 191. Depending on which position the question asks for, each successive number in row 20 follows consecutively: 191, 192, 193, etc.</p><p>∴ Answer: A</p>
Correct Answer: A