Identifying the nth Term of the Sequence 3, 12, 27, 48 and Solving Related Problems

Understanding the nth term of a sequence is crucial in many aspects of mathematics, particularly in fields that involve series and sequences. In this article, we will explore the sequence 3, 12, 27, 48, and identify the formula for its nth term. We will also cover related problems such as finding the greatest common factor (GCF) of numbers and connecting the sequence with perfect squares.

Understanding the Sequence and Its Pattern

To find the nth term of the given sequence 3, 12, 27, 48, we start by examining its pattern. Each term in the sequence can be expressed as a multiple of perfect squares, specifically:

a_1 3 times 1^2 3

a_2 3 times 2^2 12

a_3 3 times 3^2 27

a_4 3 times 4^2 48

The general formula for the nth term can be expressed as:

a_n 3n^2

Deriving the Formula Using Differences

Another way to find the nth term is through the analysis of differences between the terms:

We calculate the first differences:

12 - 3 9

27 - 12 15

48 - 27 21

Next, we calculate the second differences:

15 - 9 6

21 - 15 6

Since the second differences are constant, the sequence is a quadratic function of the form:

a_n An^2 Bn C

We set up a system of equations using the known terms:

1^2A 1B C 3 rarr; A B C 3 (1)

2^2A 2B C 12 rarr; 4A 2B C 12 (2)

3^2A 3B C 27 rarr; 9A 3B C 27 (3)

Solving the system of equations:

From (1): C 3 - A - B

Substituting C in (2) and (3):

From (2):

4A 2B 3 - A - B 12 rarr; 3A B 9 (4)

From (3):

9A 3B 3 - A - B 27 rarr; 8A 2B 24 rarr; 4A B 12 (5)

Subtracting (4) from (5):

4A B - (3A B) 12 - 9 rarr; A 3

Substituting A 3 back into (4):

3(3) B 9 rarr; 9 B 9 rarr; B 0

Substituting A 3 and B 0 back into (1):

3 0 C 3 rarr; C 0

Thus, the nth term is:

a_n 3n^2

Solving Related Problems: Greatest Common Factor (GCF)

Next, we tackle a related problem involving the greatest common factor (GCF) of the terms in the sequence:

3 1 times 3

12 1 times 12 1 times 2 times 6 1 times 3 times 4

27 1 times 27 1 times 3 times 9 1 times 3 times 3 times 3

48 1 times 48 1 times 2 times 24 1 times 3 times 16 1 times 3 times 4 times 4

From these factorizations, the GCF for each term is:

GCF(3) 3

GCF(12) 3, GCF(27) 3, GCF(48) 3

Please note that for the sequence 3, 12, 27, 48, the GCF is consistently 3.

Conclusion

In conclusion, we have identified the nth term of the sequence 3, 12, 27, 48 as a_n 3n^2. Additionally, we have explored related problems involving perfect squares and greatest common factors, providing a comprehensive understanding of the sequence and its properties.