Understanding Geometric Progressions: Finding the First Term, Common Ratio, and Seventh Term

Geometric progressions (GP) are sequences of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This article will guide you through the process of finding the first term, common ratio, and the seventh term given specific terms in a GP.

Identifying the Common Ratio

When faced with a problem involving a geometric progression, such as the third term being 1/4 and the sixth term being 1/32, a mathematician might immediately start thinking about powers of 2. In this case, the common ratio r 1/2. This means that each term is half of the previous term.

To verify this, we can observe that multiplying the third term, 1/4, by the common ratio, 1/2, twice will lead us to the sixth term, 1/32:

First term: ( a_1 ) Third term: ( a_3 a_1 cdot (r^2) 1/4 ) Sixth term: ( a_6 a_1 cdot (r^5) 1/32 )

Let's reverse the process to find the first term and common ratio more explicitly.

Verifying the Common Ratio and Finding the First Term

Given:

Third term, ( T_3 a cdot r^2 1/4 ) Sixth term, ( T_6 a cdot r^5 1/32 )

To find the common ratio ( r ), we can use the formula for the sixth term and the third term:

( T_6 / T_3 (a cdot r^5) / (a cdot r^2) 1/32 / 1/4 r^3 1/8 )

Solving for ( r ): ( r (1/8)^{1/3} 1/2 )

Now, we can use the third term to find the first term ( a ):

( a cdot r^2 1/4 ) ( a cdot (1/2)^2 1/4 ) ( a cdot 1/4 1/4 ) ( a 1 )

With the first term ( a 1 ) and the common ratio ( r 1/2 ), we can now find the seventh term:

( T_7 a cdot r^6 1 cdot (1/2)^6 1 cdot 1/64 1/64 )

Understanding and Applying Powers of 2

To deepen your understanding of geometric progressions, it is helpful to know the powers of 2. Here are some key values:

2^5 32 2^6 64 2^7 128 2^8 256 2^9 512 2^10 1024

Knowing these powers will help you quickly identify common ratios and find terms in a geometric progression.

Conclusion

Understanding geometric progressions is crucial for many fields, including mathematics, computer science, and engineering. Mastering the techniques to find the first term, common ratio, and other terms in a sequence can be applied in various scenarios, from financial modeling to signal processing.

Knowing the common ratio and the first term allows us to easily determine other terms in the sequence, as demonstrated in this article. By learning and recognizing patterns in powers of 2, you can efficiently solve problems involving geometric progressions.