What is the Common Ratio (r) in a Geometric Sequence?

In this article, we will explore the problem of finding the common ratio (r) in a geometric sequence, given that the first two terms of an arithmetic and geometric sequence are the same, and the first term of both sequences is 12. Through a detailed step-by-step analysis, we will solve the equation and find the possible values for r.

Setting Up the Problem

Let us denote the first term of both the arithmetic and geometric sequences as a 12. We need to find the common ratio (r) in the geometric sequence given certain conditions.

Arithmetic Sequence

First term: a 12 Second term: a d 12 d Third term: a 2d 12 2d

The sum of the first three terms of the arithmetic sequence is:

Sa 12 (12 d) (12 2d) 36 3d

Geometric Sequence

First term: a 12 Second term: ar 12r Third term: ar2 12r2

The sum of the first three terms of the geometric sequence is:

Sg 12 12r 12r2 12(1 r r2)

The Given Condition

According to the problem, the sum of the first three terms of the geometric sequence is 3 more than the sum of the first three terms of the arithmetic sequence:

Sg Sa 3

Substituting the expressions we found:

12(1 r r2) 36 3d 3

This simplifies to:

12(1 r r2) 39 3d

Finding d

We need to express d in terms of r. Since d is the common difference of the arithmetic sequence, we can derive d from the relationship between the two sequences. From the first two terms being equal, we can set:

12 d 12r

This implies:

d 12r - 12

Substituting d back into the equation, we get:

12(1 r r2) 39 3(12r - 12)

Expanding the right side:

12(1 r r2) 39 36r - 36

This simplifies to:

12(1 r r2) 3 36r

Expanding the left side:

12 12r 12r2 3 36r

Rearranging gives:

12r2 - 24r 9 0

Solving the Quadratic Equation

We can solve the quadratic equation by dividing the entire equation by 3:

4r2 - 8r 3 0

Using the quadratic formula, we get:

r frac{-(-8) pm sqrt{(-8)^2 - 4 cdot 4 cdot 3}}{2 cdot 4}

Calculating the discriminant:

r frac{8 pm sqrt{64 - 48}}{8} frac{8 pm sqrt{16}}{8} frac{8 pm 4}{8}

This gives us two potential values for r:

r frac{12}{8} frac{3}{2} quad or quad r frac{4}{8} frac{1}{2}

Conclusion

Thus, the values of r are:

r frac{3}{2} r frac{1}{2}