Certainly! Let's solve this step-by-step:
### Step 1: Understand the Problem
We are given:
- The first term of the geometric progression (G.P.), [tex]\( a \)[/tex], is 5.
- The sum of the first four terms of the G.P., [tex]\( S_4 \)[/tex], is 780.
We need to determine the common ratio [tex]\( r \)[/tex].
### Step 2: Write the Formula for the Sum of the First [tex]\( n \)[/tex] Terms of a G.P.
The sum of the first [tex]\( n \)[/tex] terms of a G.P. with the first term [tex]\( a \)[/tex] and common ratio [tex]\( r \)[/tex] is given by:
[tex]\[ S_n = a \left(\frac{1 - r^n}{1 - r}\right) \][/tex]
Given [tex]\( n = 4 \)[/tex], [tex]\( S_4 = 780 \)[/tex], and [tex]\( a = 5 \)[/tex], we can write:
[tex]\[ S_4 = 5 \left(\frac{1 - r^4}{1 - r}\right) \][/tex]
### Step 3: Substitute the Given Values into the Formula
Substitute [tex]\( a = 5 \)[/tex] and [tex]\( S_4 = 780 \)[/tex]:
[tex]\[ 780 = 5 \left(\frac{1 - r^4}{1 - r}\right) \][/tex]
### Step 4: Simplify the Equation
Multiply both sides by [tex]\( (1 - r) \)[/tex] to clear the fraction:
[tex]\[ 780 (1 - r) = 5 (1 - r^4) \][/tex]
[tex]\[ 780 - 780r = 5 - 5r^4 \][/tex]
### Step 5: Rearrange the Equation
Collect all terms on one side of the equation:
[tex]\[ 780 - 5 = 780r - 5r^4 \][/tex]
[tex]\[ 775 = 780r - 5r^4 \][/tex]
### Step 6: Solve the Polynomial Equation
We need to solve the equation:
[tex]\[ 775 = 780r - 5r^4 \][/tex]
This is a fourth-degree polynomial equation in terms of [tex]\( r \)[/tex]:
Reorganize to:
[tex]\[ 5r^4 - 780r + 775 = 0 \][/tex]
### Step 7: Find the Roots of the Polynomial
The solutions to this polynomial equation are the values of the common ratio [tex]\( r \)[/tex]. The solutions are:
[tex]\[ r = 1, 5, -3 - \sqrt{22}i, -3 + \sqrt{22}i \][/tex]
These are the values of [tex]\( r \)[/tex] that satisfy the given conditions:
- [tex]\( r = 1 \)[/tex]
- [tex]\( r = 5 \)[/tex]
- Two complex numbers: [tex]\( -3 - \sqrt{22}i \)[/tex] and [tex]\( -3 + \sqrt{22}i \)[/tex]
### Step 8: Interpret the Results
Given the context of a geometric progression with real terms, we are most interested in the real common ratios:
- [tex]\( r = 1 \)[/tex]
- [tex]\( r = 5 \)[/tex]
The two complex solutions [tex]\( -3 - \sqrt{22}i \)[/tex] and [tex]\( -3 + \sqrt{22}i \)[/tex] are generally not applicable in the context of a simple geometric progression with real terms.
Therefore, the possible common ratios for the G.P. are:
[tex]\[ \boxed{1 \text{ and } 5} \][/tex]
Both common ratios yield sums and terms that match the given conditions of the problem.
