Integer Sums
The sum of the positive odd integers less than 90 is subtracted from the sum of the positive even integers less than or equal to 90. What is the resulting difference?Answer
With problems like these, we need to consider the difference per each group of 2 terms
Display the first 90 even terms
Sum of the first 90 even terms = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 + 54 + 56 + 58 + 60 + 62 + 64 + 66 + 68 + 70 + 72 + 74 + 76 + 78 + 80 + 82 + 84 + 86 + 88 + 90Display the first 90 odd terms
Sum of the first 90 odd terms = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 + 35 + 37 + 39 + 41 + 43 + 45 + 47 + 49 + 51 + 53 + 55 + 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 + 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89The difference of the first 90 even terms minus the first 90 odd terms can be written as follows:
Group each set of odd and even terms
(2 - 1) + (4 - 3) + (6 - 5) + (8 - 7) + (10 - 9) + (12 - 11) + (14 - 13) + (16 - 15) + (18 - 17) + (20 - 19) + (22 - 21) + (24 - 23) + (26 - 25) + (28 - 27) + (30 - 29) + (32 - 31) + (34 - 33) + (36 - 35) + (38 - 37) + (40 - 39) + (42 - 41) + (44 - 43) + (46 - 45) + (48 - 47) + (50 - 49) + (52 - 51) + (54 - 53) + (56 - 55) + (58 - 57) + (60 - 59) + (62 - 61) + (64 - 63) + (66 - 65) + (68 - 67) + (70 - 69) + (72 - 71) + (74 - 73) + (76 - 75) + (78 - 77) + (80 - 79) + (82 - 81) + (84 - 83) + (86 - 85) + (88 - 87) + (90 - 89)Simplify our term grouping
Since each term in parentheses equals 1, we have1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Since we have 90 numbers, we have exactly ½ x 90 = 45 odd/even pairs
45 odd/even pairs x 1 = 45
On the exam, use this shortcut instead of listing out all terms, and you will get to your answer with lightning quickness.
