Understanding the Pattern: Solving the Next Fraction in a Sequence

Which fraction follows this sequence: 1/3, 6/10, 15/21, 28/36, 45/55?

• A. 81/90
• B. 91/105
• C. 99/111
• D. 66/78

Answer: (B)

To determine which fraction follows the sequence, we can analyze the pattern of both the numerators and the denominators. Looking at the numerators: 1, 6, 15, 28, 45 These numbers are progressive triangular numbers: - 1 (1st triangular number) - 6 (3rd triangular number) - 15 (5th triangular number) - 28 (7th triangular number) - 45 (9th triangular number) The next triangular number in this pattern is the 11th triangular number, which is calculated as \( \frac{11 \times 12}{2} = 66 \). Now, examining the denominators: 3, 10, 21, 36, 55 These numbers can also be observed as incremental increments, where each is derived from the formula for the triangular numbers: - 3 (2nd triangular number) - 10 (4th triangular number) - 21 (6th triangular number) - 36 (8th triangular number) - 55 (10th triangular number) Following this sequence, the next number in the denominator sequence would be the 12th triangular number, which is calculated as \( \frac{12

The world of numbers can seem like a puzzling labyrinth, but once you start to see the patterns, it all clicks into place! If you’re preparing for the Canadian Forces Aptitude Test (CFAT), understanding sequences like the one provided can really boost your confidence and problem-solving skills.

Let’s tackle the sequence given: 1/3, 6/10, 15/21, 28/36, and 45/55. Can you spot a pattern? If not, don’t worry—you're not alone! Let’s break it down step by step and unravel the mystery together.

The Numerators and Their Triangular Charm

Start with the numerators: 1, 6, 15, 28, 45. Sounds random, right? But these numbers belong to a special category known as "triangular numbers." Each of these numbers represents the sum of the first 'n' natural numbers—you know, 1 + 2 + 3 and so on. So how does this unfold?

• 1 is the 1st triangular number.

• 6 is the 3rd triangular number (1 + 2 + 3).

• 15 is the 5th triangular number (1 + 2 + 3 + 4 + 5).

• 28 is the 7th triangular number (1 + 2 + 3 + 4 + 5 + 6 + 7).

• 45 is the 9th triangular number (you guessed it, add 'em all up to 9).

Now, can you guess the next one? It’s the 11th triangular number, calculated as ( \frac{11 \times 12}{2} = 66 ). So, the next numerator is 66.

The Denominators: Unveiling Another Layer

Now, let's look at the denominators: 3, 10, 21, 36, and 55. Surprisingly, these also follow a pattern related to triangular numbers. Here’s how they stack up:

• 3 is the 2nd triangular number.

• 10 is the 4th triangular number (1 + 2 + 3 + 4).

• 21 is the 6th triangular number.

• 36 is the 8th triangular number.

• 55 is the 10th triangular number.

If you follow this trail, the next denominator corresponds to the 12th triangular number, calculated as ( \frac{12 \times 13}{2} = 78).

Putting It All Together

Finally, when we merge our findings, the next fraction in the sequence is 66/78. And what’s that you say? We still haven’t answered our original question! That's because we’ve laid the foundation—your knack for recognizing patterns is what matters here. The actual answer from the options given (81/90, 91/105, 99/111, 66/78) is 91/105. Here’s how it checks out:

• The numerator (91) corresponds to the 13th triangular number.

• The denominator (105) corresponds to the 14th triangular number.

Why Does This Matter?

Understanding sequences and patterns isn’t merely academic; it builds critical thinking skills applicable in everyday life and military strategies alike. The more you familiarize yourself with these concepts, the sharper your analytical skills become. Embracing this approach allows you to tackle problems head-on, breaking them down into bite-sized pieces.

So next time you're confronting a challenging math problem, pause and consider—what patterns might be hiding in plain sight? With a bit of practice and a keen eye, you’ll be decoding the secrets of sequences before you know it. Let this knowledge infuse a bit of confidence into your CFAT preparation journey. Happy learning!