Determine the Common Ratio in a Geometric Sequence Using Given Terms

In this guide, we will explore how to determine the common ratio, ( r ), in a geometric sequence given the second and fifth terms. A geometric sequence is a sequence 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.

The ( n )-th term of a geometric sequence can be expressed as:

$ a_n a_1 cdot r^{n-1} $

Given Information

We are given the second term, ( a_2 24 ), and the fifth term, ( a_5 1536 ).

Setting Up Equations

Using the general formula for a geometric sequence, we can express the second and fifth terms as follows:

Second Term

$ a_2 a_1 cdot r^{2-1} a_1 cdot r 24 $

Fifth Term

$ a_5 a_1 cdot r^{5-1} a_1 cdot r^4 1536 $

Solving for the Common Ratio

First, we solve the second equation for ( a_1 ):

$ a_1 frac{24}{r} $

Substitute this expression into the equation for the fifth term:

$ left( frac{24}{r} right) cdot r^4 1536 $

Which simplifies to:

$ 24r^3 1536 $

Divide both sides by 24:

$ r^3 frac{1536}{24} 64 $

Take the cube root of both sides:

$ r sqrt[3]{64} 4 $

Conclusion

The common ratio ( r ) is:

$ boxed{4} $

Verification

To verify, we can substitute ( r 4 ) back into the equation for the second term:

$ a_2 a_1 cdot 4 24 $

Solving for ( a_1 ):

$ a_1 frac{24}{4} 6 $

Now, check the fifth term:

$ a_5 6 cdot 4^4 6 cdot 256 1536 $

This confirms our solution.

Bonus: Deriving the Sequence

Given the common ratio ( r 4 ), we can derive the first few terms of the sequence:

1st term: ( a_1 6 ) 2nd term: ( a_2 6 cdot 4 24 ) 3rd term: ( a_3 24 cdot 4 96 ) 4th term: ( a_4 96 cdot 4 384 ) 5th term: ( a_5 384 cdot 4 1536 )

This confirms the sequence:

6, 24, 96, 384, 1536, 6144