<p>Find the total two-digit numbers formed using the expression involving \( \displaystyle\sum_{r=2}^{13}(7r+2) \) and \( \displaystyle\sum_{r=1}^{13}(7r+5) \).</p>
Step-by-Step Solution
Key Concept: Calculate both arithmetic series sums separately using the summation formula, then combine them to find two-digit numbers formed. The sum ∑(7r+2) and ∑(7r+5) are arithmetic series that can be split into linear and constant parts.
<p><strong>Step 1: Calculate ∑(7r+2) for r=2 to 13</strong></p><p>∑(7r+2) = 7∑r + 2(number of terms)</p><p>Number of terms = 13 - 2 + 1 = 12</p><p>∑r from r=2 to 13 = ∑r from r=1 to 13 - 1 = 13(14)/2 - 1 = 91 - 1 = 90</p><p>∑(7r+2) = 7(90) + 2(12) = 630 + 24 = 654</p><p><strong>Step 2: Calculate ∑(7r+5) for r=1 to 13</strong></p><p>∑(7r+5) = 7∑r + 5(number of terms)</p><p>Number of terms = 13</p><p>∑r from r=1 to 13 = 13(14)/2 = 91</p><p>∑(7r+5) = 7(91) + 5(13) = 637 + 65 = 702</p><p><strong>Step 3: Find two-digit numbers formed</strong></p><p>The two-digit numbers that can be formed from the digits in 654 and 702 are: 65, 54, 70, 72, 02(invalid)</p><p>Valid two-digit numbers: 65, 54, 70, 72, and their permutations/combinations</p><p>Sum of all two-digit numbers = 654 + 702 = 1356</p><p>∴ Answer: <strong>1356</strong></p>
Correct Answer: 1356