In this article, we are going to look at a specific type of question commonly asked in various competitive exams. Have a look at these two examples:
Example 1: All the page numbers from a book are added, beginning at page 1. However, one page number was mistakenly added twice. The sum obtained was 1000. Which page number was added twice?
Example 2: All the page numbers from a book are added, beginning at page 1. However, one page number was missed during addition. The sum obtained was 360. Which page number was missed?
These questions of the format ‘a number added twice or a number missed while adding the first n natural numbers’ are very easy to solve and can be answered in three simple steps. Let’s understand how we can answer these questions.
Let’s start with example 1:
All the page numbers from a book are added, beginning at page 1. However, one page number was mistakenly added twice. The sum obtained was 1000. Which page number was added twice?
Solution: To solve this question, we are basically looking at sum of first n natural numbers with a specific number x (1 ≤ x ≤ n) added twice which can be written as:
From here, follow these steps to arrive at your answer without even touching the paper (yes, this can be solved without writing anything)
Step 1: Multiply the sum by 2.
Step 2: Take the square root of this product. Consider the integral part only.
Step 3. This will give you ‘n’. Substitute this in the formula to get x.
Applying this to the above question:
Step 1: 1000 × 2 = 2000
Step 2: Square root of 2000 = 44 {you might think this is difficult but it is not. We know square of 40 is 1600 and 50 square is 2500. 2000 lies somewhere in between so look for square of 45 which is nothing but 2025. Hence, square root of 2000 will be 44.something}
Step 3: That gives us n = 44. Substitute this in our formula to get x
The reason why we take square root of the product is because n (n+1) is going to be pretty close to square of n. {(n)(n+1) = n^2 + n)}
1000 = 44(45)/2 + x
1000 = 990 + x
x = 10
In other words, the book was of 44 pages and someone was adding 1 + 2 + … + 44. The sum should have been 990 but it was reported 1000. Clearly, 10 was added twice.
Let’s move to example 2:
All the page numbers from a book are added, beginning at page 1. However, one page number was missed during addition. The sum obtained was 360. Which page number was not added to the sum?
Solution: Unlike the first question, this time the number is not getting added even once while taking the summation. Hence, the formula will look like:
If we follow the steps mentioned in the earlier solution,
Step 1: 360 × 2 = 720
Step 2: Square root of 720 =27 {We know the square of 27 = 729}
Step 3: That gives us n = 27. Substitute this in our formula to get x
360 = 27(28)/2 – x
360 = 378 – x
x = 18
In other words, the book was of 27 pages and someone was adding 1 + 2 + … + 27. The sum should have been 378 but it was reported 360. This difference was because 18 wasn’t added while doing this exercise.
Isn’t that simple to do? So the next time you see a question like this, answer it using these three steps. Let us know your feedback on this article and stay tuned for more…
