Understanding the Common Ratio in a Geometric Progression

A geometric progression (GP) 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. This concept is fundamental in many areas of mathematics and has practical applications in various fields, including finance and engineering.

The Problem and Solution

Given a geometric progression (GP) with the first term a 2, the sum of the first four terms equals 30, and the sum of the last four terms equals 960, we need to find the common ratio r.

Step 1: Sum of the First Four Terms

The first four terms of the GP can be expressed as:

The sum of these terms is given as 30:

S4 a ar ar2 ar3 30

Substituting a 2, we get:

2 2r 2r2 2r3 30

Dividing through by 2:

1 r r2 r3 15

Step 2: Sum of the Last Four Terms

Let's denote the total number of terms in the GP as n. The last four terms are:

The sum of these terms is given as 960:

arn-4 arn-3 arn-2 arn-1 960

Factoring out arn-4 from the sum:

arn-4 (1 r r2 r3) 960

From Step 1, we know that 1 r r2 r3 15:

arn-4 times; 15 960

Solving for rn-4:

rn-4 frac{960}{30} 32

Step 3: Relating r and n

Since 32 25, we have:

rn-4 25

This implies that r 2 and n-4 5:

r 2 AND n 9

Step 4: Finding the Common Ratio r

Finally, substituting r 2 into the equation 1 r r2 r3 15:

1 2 22 23 1 2 4 8 15

As expected, this satisfies the given condition. Therefore, the common ratio r is:

boxed{2}

Conclusion

The common ratio r of the geometric progression, given the conditions, is 2. This example demonstrates the steps required to solve for the common ratio in a geometric progression, illustrating the method and ensuring a clear understanding of the process.