Geometric Progression in the Sequence 2, 6, 18, 54, 162...: Understanding and Calculating the Ratio of the 51st Term to the 48th Term

In mathematics, especially in the realm of sequences and series, a geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. This article discusses how to determine the value of the ratio of the 51st term to the 48th term in a specific geometric sequence, namely 2, 6, 18, 54, 162…

Understanding the Sequence

The given sequence is a geometric progression (GP) where each term is a multiple of the previous term by a factor of 3. The first term (a1) of this sequence is 2, and the common ratio (r) is 3.

Formulating the General Term of a Geometric Progression

The general term formula for a geometric sequence is given by:

(a_n a_1 cdot r^{n-1})

For the given sequence, we can express the n-th term as:

(a_n 2 cdot 3^{n-1})

Calculating Specific Terms

To find the 48th term (A) and the 51st term (B), we apply the general formula:

(A a_{48} 2 cdot 3^{48-1} 2 cdot 3^{47})

(B a_{51} 2 cdot 3^{51-1} 2 cdot 3^{50})

Calculating the Ratio B/A

To find the ratio of the 51st term to the 48th term (B/A), we divide B by A:

(frac{B}{A} frac{2 cdot 3^{50}}{2 cdot 3^{47}} 3^{50-47} 3^3 27)

Therefore, the value of (frac{B}{A}) is 27.

Alternative Approaches

Another way to solve this problem is to recognize that the common ratio (r) is 3. Since we are dealing with terms that are 3 positions apart in the sequence, we can calculate the ratio by raising the common ratio to the appropriate power:

(r^3 3^3 27)

Using the same logic, we can confirm this by:

Assuming the 48th term is 54 and the 51st term is 162:

(frac{162}{54} frac{2 cdot 3^{50}}{2 cdot 3^{47}} 3^3 27)

Assuming the 48th term is 6 and the 51st term is 18:

(frac{18}{6} frac{2 cdot 3^5}{2 cdot 3^2} 3^3 27)

Conclusion

In summary, the value of the ratio of the 51st term to the 48th term in the sequence 2, 6, 18, 54, 162… is 27. This solution is derived using the general formula for geometric sequences and the property of the common ratio in a GP.

Keywords: geometric sequence, ratio, common ratio, geometric progression