Questions that involve the summation formula, whether on their own or one component of a more complicated problem, often trip test-takers up for the simplest of reasons: figuring out “how many items” are in the set can sometimes prove tricky. One way to avoid the headache of trying to remember the rule for each different kind of limitation (consecutive even/odd/other, inclusive vs. exclusive, whether the set starts/ends with an even/odd), is to simply employ a strategy that will quickly and consistently allow you to determine the number of items in the set: patterns.
Before we delve into how, let’s review the summation formula and when it’s used. The summation formula:
∑ = (# of Numbers in the Set)(Largest Number + Smallest Number)/2
The formula is used to quickly calculate the sum of a relatively large set of consecutive integers. It is important to note that the formula only works for consecutive numbers, and that you need to be able to determine the total number of numbers for which you’ll be finding the sum.
That calculation, the number of numbers in the set, is usually the trickiest part. I’ve seen students try to memorize a specific rule for each different kind of permutation that can arise. There’s a better, more consistent way:
- Use a much smaller, though representative set.
- Count the number of items that would be included based on the parameters of the problem.
- Create a simple formula or equation using the first and last numbers in your small set (usually some combination
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of subtraction, division, and addition/subtraction) that can be applied to the larger set.
- Apply to larger set.
Let’s use an example to see how it would work:
If p is the sum of all the even integers between 91 and 499, what is p?
To solve this question using the summation formula we would need to know how many even integers there are between 91 and 499.
To determine this we’ll figure out a quick, easy formula using a smaller representative set.
- The parameters in the question are two odd numbers. So, let’s say 1 to 9.
- We need all the even numbers, which would be 2, 4, 6, and 8. So 4 numbers in total.
- Using the end numbers 1 and 9 we could take the difference and divide by two to get the number of numbers in the set. 9 – 1 = 8 8/2 = 4.
- Doing the same with the original end numbers we get 499 – 91 = 408 408/2 = 204. Thus, our original set has 204 numbers in it. We can now use this in the summation formula to determine the sum.
This method is effective for any situation whereby you need to find the number of numbers in a consecutive set of numbers. For example, in a problem that requires you to find the probability of selecting a number from a set of consecutive numbers:
Raffle tickets numbered consecutively from 60 to 405 have been put into a bin for a raffle. What is the probability that the number of the ticket selected will be divisible by two?
In order to find probability, we need the number of items that satisfy the condition (divisible by 2; i.e. even), and the total number of items. To find the number of evens, you could count them (Not!) or you could use the strategy we used above (Ding-ding-ding!).
Patterns are one of the keys to GMAT success. Developing efficient, applicable methods to expedite calculations or avoid tedious memorization also help to sharpen your higher order thinking skills, the hallmark of a high scorer.
Best of luck!
If you think your higher order thinking skills could use a boost, consider signing up for a course or tutoring with us. Our materials and instruction not only teach you the rules you need to know but help you recognize when and how to use them!