What is the Sum of the First 100 Positive Odd Numbers? GMAT Problem Solving

Question: What is the sum of the first 100 positive odd numbers?

1. 5,000
2. 7,500
3. 8,000
4. 10,000
5. 12,000

“What is the Sum of the First 100 Positive Odd Numbers?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Graduate Management Admission Test Exam Practice”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:
Approach Solution 1:
This one is about pattern recognition. Set up a table as follows:

Number Sum
1 1
3 4
5 9
7 16
9 25

Now as you see, the sum always equals the perfect square of the number of odd integers that you want to sum. So the sum of the first 5 odd integers is 5^2 = 25. The pattern continues. Thus, the sum of the first 100 odd integers will be equal to 100^2 = 10,000.

Correct Answer: D

Approach Solution 2:
Sum of the first 2 odd numbers = 1 + 3 = 4
Sum of the first 3 odd numbers = 1 + 3 + 5 = 9
Sum of the first 4 odd numbers = 1 + 3 + 5 + 7 = 16
Sum of the first 5 odd numbers = 1 + 3 + 5 + 7 + 9 = 25

Notice that:
Sum of the first 2 odd numbers = 2²
Sum of the first 3 odd numbers = 3²
Sum of the first 4 odd numbers = 4²
Sum of the first 5 odd numbers = 5²

In general, the sum of the first n odd numbers = n²
So, the sum of the first 100 odd numbers = 100² = 10,000

Correct Answer: D

Approach Solution 3:
Formula: 1 + 2 + 3 . . . + n = (n)(n + 1)/2

Let's use the above formula to find the sum of the first 200 integers (including odds AND evens)

1 + 2 + 3 . . . + 199 + 200 = (200)(200 + 1)/2

= (200)(201)/2
= (100)(201)
= 20,100

So, the sum of the first 200 integers is 20,100

HALF of those integers are ODD and HALF are even. So, this sum includes the sum of the first 100 ODD integers and the sum of the first 100 EVEN integers.

So, the sum of the first 100 ODD integers is APPROXIMATELY 20,100/2
20,100/2 = 10,050
So, the sum of the first 100 ODD integers = 10,050
Answer choice D is the only one that's close to 10,050 so it must be the correct answer.

Correct Answer: D

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