What is the sum of the first 50 positive integers?
This is a daunting question, perhaps akin to the question that asks about two trains traveling at varying rates headed in the same direction. Just like the speeding locomotive question, this type of question should not intimidate you – once you understand the concept.
Think of it this Way
If I add up the first number and last number, I get 1 + 50 = 51. If I begin working my way inwards, i.e. imagining a number line, I next add 49 + 2 = 51, then 48 + 3 = 51. You will notice that the sum is always 51. So, as long as I increase by one at the low range, and decrease by 1 from the upper range, the sum of the two numbers will always equal 51.
The next question I want to ask myself is, how many pairs of numbers are there in the first 50 integers? The logic is, if we pair numbers the way we did in the preceding paragraph, we always get 51. So, I’m asking myself, how many 51’s are there. Dividing 50 by 2, we get the number of pairs: 25. Therefore, we have to multiply 25 x 51 to get the sum of the sequence, which is 1275.
So, whenever we need to find a consecutive series, we simply add the first plus the last (e.g. 1 + 50), and then take the number of digits (e.g. 50) and divide by 2 (remember we are looking for the pairs). Next, I multiply this result (50/2) by the first and last (1 + 50), and I get 25 x 51 = 1275.
Let’s try that with another, easier problem: What is the sum of the numbers 1 – 10.
Adding first plus last (1 + 10), I get eleven. Then, I take the number of digits, 10, divide by 2, and get 5. Next, I simply multiply 11 x 5 = 55.
Time for a Word Problem
Bob is training for a fitness competition. In order to increase his maximum number of pull-ups, he follows the following routine: he begins with 25 pull-ups, rests for thirty seconds, and then does 24 pull-ups and rests, dropping one pull-up each time (25, 24, 23, etc.) until his final set of 11 pull-ups. How many total pull-ups does Bob do?
You could furiously add up the numbers 25 through 11, or, you could find the sum of all the numbers 1 – 25, and then subtract from that the sum of 1 to 10 (remember, Bob stops at 11 pull-ups). To find the sum of 1 – 25, we add first + last = 26 x 25/2 = 325. Now, we subtract the sum of 1 – 10, which is 55 (see above). This gives us a total of 270 pull-ups (Bob is obviously pretty strong).
The best way to improve at sequences is practice. At the same time, there are other twists to sequences. In the coming week, I will deal with more advanced problems, and other helpful tricks to help you deal with a type of problem that should not be intimidating. If you’d like a review of the basics, read GRE Math Video Lessons: Series and Counting Basics to make sure you have a solid grasp of every level.