Introduction to Geometric Sequences

Geometric sequences are a fascinating area of mathematics where each term is multiplied by a constant ratio to get the next term. In this article, we will explore how to find the value of k so that the terms 3k 1, 3k 4, and 7k-6 form a geometric sequence. We will break down the problem step-by-step, using algebraic manipulation and common ratio principles.

Understanding the Problem

For the terms 3k 1, 3k 4, and 7k-6 to form a geometric sequence, the ratio between consecutive terms must be constant. This means:

(frac{3k4}{3k1} frac{7k6}{3k4})

Let's solve this equation to find the possible values of k that satisfy this condition.

Solving the Equation

Starting with the given equation:

(frac{3k4}{3k1} frac{7k6}{3k4})

We can cross-multiply to get:

((3k 4)^2 (3k 1)(7k-6))

Expanding both sides:

(9k^2 24k 16 21k^2 - 6k 7k - 6)

Simplifying the equation:

(9k^2 24k 16 21k^2 k - 6)

Bringing all terms to one side:

(0 12k^2 - 25k - 22)

Dividing through by 12, we get:

(0 k^2 - frac{25}{12}k - frac{11}{6})

Solving this quadratic equation using the quadratic formula:

(k frac{-b pm sqrt{b^2 - 4ac}}{2a})

Here, (a 1), (b -frac{25}{12}), and (c -frac{11}{6}). Plugging these values into the formula:

(k frac{frac{25}{12} pm sqrt{left(frac{25}{12}right)^2 - 4 cdot 1 cdot left(-frac{11}{6}right)}}{2 cdot 1})

Simplifying the expression inside the square root:

(k frac{frac{25}{12} pm sqrt{frac{625}{144} frac{44}{6}}}{2})

Simplifying further:

(k frac{frac{25}{12} pm sqrt{frac{625}{144} frac{1056}{144}}}{2})

(k frac{frac{25}{12} pm sqrt{frac{1681}{144}}}{2})

(k frac{frac{25}{12} pm frac{41}{12}}{2})

Thus:

(k frac{25 41}{24} frac{66}{24} frac{11}{4}) or (k frac{25 - 41}{24} frac{-16}{24} -frac{2}{3})

So the possible values for k are (k frac{11}{4}) or (k -frac{2}{3}).

Conclusion

The value of (k) that satisfies the condition for the terms 3k 1, 3k 4, and 7k-6 to form a geometric sequence is (k frac{11}{4}) or (k -frac{2}{3}).

Practice Questions

Try practicing similar problems to deepen your understanding of geometric sequences and algebraic manipulation.

Keywords

geometric sequence, common ratio, algebraic manipulation