Understanding Geometric Sequences and Finding the Seventh Term

Welcome to this detailed guide on understanding and solving geometric sequences, particularly focusing on finding the seventh term of a given series. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Definition and Characteristics of Geometric Sequences

A geometric sequence is defined by the relationship where each term is the product of the previous term and a constant multiplier, known as the common ratio. The nth term of a geometric sequence can be calculated using the formula:

nth Term of a Geometric Sequence (GP)

T[n] a * r^(n-1)

a: First term of the sequence r: Common ratio n: Term number

Given Sequence and Its Key Characteristics

The sequence provided is 1, -3, 9, -27, 81, -243, 729. Let's break down how this sequence is composed:

Identifying the Common Ratio

By observing the sequence, we can identify the common ratio, which is -3:

T[2]/T[1] -3/1 -3

T[3]/T[2] 9/-3 -3

T[4]/T[3] -27/9 -3

T[5]/T[4] 81/-27 -3

T[6]/T[5] -243/81 -3

T[7]/T[6] 729/-243 -3

Calculating the Seventh Term

To find the seventh term in the sequence, we use the formula for the nth term of a geometric sequence:

T[7] a * r^(n-1)

a 1 (the first term) r -3 (the common ratio) n 7 (since we are looking for the seventh term)

Plugging in the values, we get:

T[7] 1 * (-3)^(7-1) 1 * (-3)^6 729

Alternative Verification

Another method to verify the seventh term is by recognizing the pattern. Starting from the first term and multiplying by -3 repeatedly:

1st term: 1 2nd term: 1 * (-3) -3 3rd term: -3 * (-3) 9 4th term: 9 * (-3) -27 5th term: -27 * (-3) 81 6th term: 81 * (-3) -243 7th term: -243 * (-3) 729

Additional Insights into Geometric Sequences

In more complex scenarios, you may need to find the sum of the first n terms of a geometric sequence. The formula for the sum of the first n terms (S[n]) is:

S[n] a * (r^n - 1) / (r - 1)

For the given sequence, the sum of the first seven terms would be:

S[7] 1 * ((-3)^7 - 1) / (-3 - 1) 1 * (-2187 - 1) / -4 1 * -2188 / -4 547

Conclusion

To summarize, the seventh term of the geometric sequence 1, -3, 9, -27, 81, -243, 729 is 729, which can be derived using the formula for the nth term of a geometric sequence and by recognizing the pattern. Understanding these concepts is crucial for solving a wide range of mathematical problems involving geometric sequences.