Introduction to Geometric Sequences

Geometric sequences are a series 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. This article will walk you through the process of finding the first term (a_1) of a geometric sequence given specific terms, focusing on the example where (a_2 8) and (a_5 -64).

Understanding the Problem

We are given a geometric sequence where:

(a_2 8),

(a_5 -64),

and we need to find the first term (a_1).

The Formula and Equations

The general formula for the (n)-th term of a geometric sequence is:

(a_n a_1 cdot r^{n-1}),

where (a_1) is the first term and (r) is the common ratio.

Step 1: Set Up the Equations

Given (a_2 8), and (a_5 -64), we can write:

(a_2 a_1 cdot r)

(a_5 a_1 cdot r^4)

Substituting the given values, we get:

(a_1 cdot r 8)

(a_1 cdot r^4 -64)

Step 2: Solve for the Common Ratio (r)

From the first equation, we can solve for (r):

(r frac{8}{a_1})

Substitute (r) into the second equation:

(a_1 cdot left(frac{8}{a_1}right)^4 -64)

This simplifies to:

(frac{8^4}{a_1^3} -64)

(frac{4096}{a_1^3} -64)

Step 3: Solve for (a_1)

Multiplying both sides by (a_1^3), we get:

4096 -64 cdot a_1^3

(a_1^3 frac{4096}{-64})

(a_1^3 -64)

Taking the cube root of both sides, we find:

(a_1 sqrt[3]{-64} -4)

Verification

We can verify the solution by finding (r) and checking that the given terms are satisfied.

From (a_2 a_1 cdot r 8):

(-4 cdot r 8) implies (r -2)

Now check (a_5):

(a_5 a_1 cdot r^4 -4 cdot (-2)^4 -4 cdot 16 -64)

Both conditions are satisfied, confirming that (a_1 -4) is correct.

Additional Methods and Insight

Another approach to solving the problem is to directly find the common ratio (r). Since moving from (a_2) to (a_5) involves multiplying by (r) three times, we can set up the following:

(a_5 a_2 cdot r^3)

(-64 8 cdot r^3)

(r^3 frac{-64}{8} -8)

(r sqrt[3]{-8} -2)

Using (a_2 8) and (r -2), we can find (a_1) as:

(a_2 a_1 cdot r)

(8 a_1 cdot (-2))

(a_1 frac{8}{-2} -4)

Conclusion

The first term (a_1) of the geometric sequence, given (a_2 8) and (a_5 -64), is (a_1 -4).