Sum of Odd Numbers 1-50?Date: 10/10/2001 at 01:09:51 From: Shaundra Subject: Problem Solving Looking for patterns What is the sum of the first 50 odd numbers? Date: 10/10/2001 at 13:40:44 From: Doctor Ian Subject: Re: Problem Solving Looking for patterns Hi Shaundra, The sum of the first 5 odd numbers is 1 + 3 + 5 + 7 + 9 = [sum] Twice this sum is (1 + 3 + 5 + 7 + 9) + (1 + 3 + 5 + 7 + 9) = [sum] * 2 We can rewrite it this way: (1 + 9) + (3 + 7) + (5 + 5) + (7 + 3) + (9 + 1) = [sum] * 2 10 + 10 + 10 + 10 + 10 = [sum] * 2 5 * 10 = [sum] * 2 (5 * 10) / 2 = [sum] = 25 All I did was match the numbers up in reverse order: _____________________________________ | _____________________________ | | | _____________________ | | | | | _____________ | | | | | | | _____ | | | | | | | | | | | | | | (1 + 3 + 5 + 7 + 9) + (1 + 3 + 5 + 7 + 9) = [sum] * 2 This is a nice trick, because each pair adds up to the same amount -- the sum of the largest and smallest numbers in the list! Can you see why it's as quick to do this for 50 numbers - or 500, or 5000 - as it is for 5? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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