Understanding the Geometric Progression Problem

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. In this problem, we are given the first term, the last term, and the sum of a geometric progression. Our goal is to find the common ratio. This involves using the properties and formulas associated with a geometric progression to determine the unique value of the common ratio that satisfies the given conditions. Finding the Geometric Progression Common Ratio is a standard problem in sequence and series.

Given Information: First Term, Last Term, and Sum

We are provided with the following details about the geometric progression:

• First term ($a$) = 7
• Last term ($l$) = 448
• Sum of the geometric progression ($S_n$) = 889

We need to use this information to find common ratio ($r$) of the sequence.

Formula for Sum of a Geometric Progression

The sum ($S_n$) of a finite geometric progression can be calculated using different formulas depending on the available information. One useful formula relates the sum, the first term, the last term, and the common ratio ($r$).

The formula for the sum of GP when the first term ($a$), last term ($l$), and common ratio ($r$) are known (and $r \neq 1$) is:

$$S_n = \frac{lr - a}{r - 1}$$

This formula is particularly helpful when we are given the first term, last term, and sum, and need to find common ratio.

Calculating the Geometric Progression Common Ratio

Now, let's use the given values and the formula $S_n = \frac{lr - a}{r - 1}$ to calculate the Geometric Progression Common Ratio. We will substitute the given values into the formula and solve for $r$.

Step-by-Step Calculation:

Substitute the given values $a=7$, $l=448$, and $S_n=889$ into the sum formula:

$$889 = \frac{(448)(r) - 7}{r - 1}$$

Multiply both sides by $(r - 1)$ to eliminate the denominator:

$$889(r - 1) = 448r - 7$$

Distribute 889 on the left side:

$$889r - 889 = 448r - 7$$

Rearrange the terms to group the 'r' terms on one side and constant terms on the other side. Subtract $448r$ from both sides and add 889 to both sides:

$$889r - 448r = 889 - 7$$

Perform the subtraction on both sides:

$$441r = 882$$

Now, isolate $r$ by dividing both sides by 441:

$$r = \frac{882}{441}$$

Perform the division:

$$r = 2$$

Thus, the calculated common ratio of the geometric progression is 2.

Verifying the Common Ratio

With $a=7$ and $r=2$, let's see the initial terms: 7, 14, 28, 56, 112, 224, 448. The last term is 448. The sum is $7+14+28+56+112+224+448 = 889$. This confirms our calculated Geometric Progression Common Ratio is correct based on the given first term, last term, and sum of GP.

Final Answer for the Common Ratio

Based on our calculation using the given first term, last term, and sum of GP, the Geometric Progression Common Ratio is 2.