How to Find the Seventh Term of a Geometric Sequence: √2 √6 3√2…
Geometric sequences are a fascinating topic in mathematics. They consist of a series 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. If you're trying to find the seventh term of a specific geometric sequence, such as √2, √6, 3√2, and so on, you'll delve into some interesting mathematical concepts. Let's break down the process and understanding of finding the seventh term of this sequence.
Understanding the Sequence
The given sequence is √2, √6, 3√2, and so on. To better understand this sequence, let's express each term in a more simplified form:
First Term (A): √2 Second Term (AR): √6 √(2*3) Third Term (AR2): 3√2 √(2*9) √(2*32)From the above, we notice that the sequence can be written as √2, √(2*3), √(2*9), and so on. Each term is obtained by multiplying the previous term by √3.
Determining the Common Ratio
As we observe the sequence, the common ratio (R) can be identified as √3. To confirm this, let's calculate the ratio between consecutive terms:
Second term divided by the first term: (√6) / (√2) √(2*3) / √2 √3 R Third term divided by the second term: (3√2) / (√6) √(2*9) / √(2*3) √(9/3) √3 RThis confirms that the common ratio R is indeed √3.
Formula for the Nth Term of a Geometric Sequence
The formula for the Nth term (TN) of a geometric sequence is given by:
TN A * R(N-1)
Here, A is the first term, and R is the common ratio. For our sequence, A √2 and R √3.
Finding the Seventh Term
To find the seventh term (T7), we will use the formula:
T7 A * R(7-1)
Substituting the values of A and R into the formula:
T7 √2 * (√3)6
Now, let's simplify this expression.
Calculate (√3)6: (√3)6 (3(1/2))6 33 27 Multiply the results: T7 √2 * 27Therefore, the seventh term T7 is:
T7 27√2Conclusion
Finding the seventh term of a geometric sequence is a straightforward process once you understand the basics of geometric sequences and their properties. The sequence given in the problem, √2, √6, 3√2, can be generalized as a geometric sequence with the first term √2 and the common ratio √3. Using the formula for the Nth term, we can easily find that the seventh term is 27√2.
Practice and Further Learning
Now that you've gone through the process, try finding the other terms of the sequence or apply similar steps to find the terms of other geometric sequences. You can also explore how geometric sequences are used in real-life applications, such as compound interest, population growth, or radioactive decay.