IDNLearn.com offers a seamless experience for finding and sharing knowledge. Join our interactive Q&A community and get reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
To find the common ratio [tex]\(r\)[/tex] of a geometric progression (G.P.) where the sum of the first eight terms is five times the sum of the first four terms, we can follow these steps:
1. Express the sums using the sum formula for a G.P.
The sum of the first [tex]\(n\)[/tex] terms of a geometric progression is given by:
[tex]\[ S_n = a \frac{1 - r^n}{1 - r} \][/tex]
where [tex]\(a\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the number of terms.
Therefore, the sum of the first eight terms ([tex]\( S_8 \)[/tex]) and the sum of the first four terms ([tex]\( S_4 \)[/tex]) can be written as:
[tex]\[ S_8 = a \frac{1 - r^8}{1 - r} \][/tex]
[tex]\[ S_4 = a \frac{1 - r^4}{1 - r} \][/tex]
2. Set up the equation according to the given condition:
We are given that the sum of the first eight terms is five times the sum of the first four terms:
[tex]\[ S_8 = 5 \cdot S_4 \][/tex]
Substituting the expressions for [tex]\( S_8 \)[/tex] and [tex]\( S_4 \)[/tex]:
[tex]\[ a \frac{1 - r^8}{1 - r} = 5 \left(a \frac{1 - r^4}{1 - r}\right) \][/tex]
3. Simplify the equation:
Since [tex]\( a \)[/tex] and [tex]\( (1 - r) \)[/tex] are common factors on both sides of the equation, they cancel out:
[tex]\[ 1 - r^8 = 5 (1 - r^4) \][/tex]
Simplifying further:
[tex]\[ 1 - r^8 = 5 - 5r^4 \][/tex]
Bring all the terms to one side to form a polynomial equation:
[tex]\[ r^8 - 5r^4 + 4 = 0 \][/tex]
4. Solve the polynomial equation by substitution:
Let [tex]\( x = r^4 \)[/tex]. This transforms the equation into a quadratic equation in terms of [tex]\( x \)[/tex]:
[tex]\[ x^2 - 5x + 4 = 0 \][/tex]
Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 4 \)[/tex]:
[tex]\[ x = \frac{5 \pm \sqrt{25 - 16}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{9}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm 3}{2} \][/tex]
Thus, the solutions are:
[tex]\[ x = 4 \quad \text{and} \quad x = 1 \][/tex]
5. Relate [tex]\( x \)[/tex] back to [tex]\( r^4 \)[/tex]:
Recall that [tex]\( x = r^4 \)[/tex].
[tex]\[ r^4 = 4 \implies r = \pm \sqrt[4]{4} = \pm \sqrt{2} \][/tex]
[tex]\[ r^4 = 1 \implies r = \pm \sqrt[4]{1} = \pm 1 \][/tex]
6. Verify the valid values of [tex]\( r \)[/tex]:
Substitute [tex]\( r = \pm 1 \)[/tex] back into the original condition [tex]\( 1 - r^8 = 5 - 5r^4 \)[/tex]. We see that:
[tex]\[ 1 - 1^8 = 5 - 5 \cdot 1 \implies 0 \neq 0 \quad (\text{This does not hold true}) \][/tex]
Therefore, [tex]\( r = \pm 1 \)[/tex] are invalid solutions.
The only valid solutions for the common ratio [tex]\( r \)[/tex] are [tex]\( \sqrt{2} \)[/tex] and [tex]\( -\sqrt{2} \)[/tex]. Therefore, the correct answers are:
(a) [tex]\( \sqrt{2} \)[/tex]
(b) [tex]\( -\sqrt{2} \)[/tex]
Hence, the correct answer is:
(c) both
1. Express the sums using the sum formula for a G.P.
The sum of the first [tex]\(n\)[/tex] terms of a geometric progression is given by:
[tex]\[ S_n = a \frac{1 - r^n}{1 - r} \][/tex]
where [tex]\(a\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the number of terms.
Therefore, the sum of the first eight terms ([tex]\( S_8 \)[/tex]) and the sum of the first four terms ([tex]\( S_4 \)[/tex]) can be written as:
[tex]\[ S_8 = a \frac{1 - r^8}{1 - r} \][/tex]
[tex]\[ S_4 = a \frac{1 - r^4}{1 - r} \][/tex]
2. Set up the equation according to the given condition:
We are given that the sum of the first eight terms is five times the sum of the first four terms:
[tex]\[ S_8 = 5 \cdot S_4 \][/tex]
Substituting the expressions for [tex]\( S_8 \)[/tex] and [tex]\( S_4 \)[/tex]:
[tex]\[ a \frac{1 - r^8}{1 - r} = 5 \left(a \frac{1 - r^4}{1 - r}\right) \][/tex]
3. Simplify the equation:
Since [tex]\( a \)[/tex] and [tex]\( (1 - r) \)[/tex] are common factors on both sides of the equation, they cancel out:
[tex]\[ 1 - r^8 = 5 (1 - r^4) \][/tex]
Simplifying further:
[tex]\[ 1 - r^8 = 5 - 5r^4 \][/tex]
Bring all the terms to one side to form a polynomial equation:
[tex]\[ r^8 - 5r^4 + 4 = 0 \][/tex]
4. Solve the polynomial equation by substitution:
Let [tex]\( x = r^4 \)[/tex]. This transforms the equation into a quadratic equation in terms of [tex]\( x \)[/tex]:
[tex]\[ x^2 - 5x + 4 = 0 \][/tex]
Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 4 \)[/tex]:
[tex]\[ x = \frac{5 \pm \sqrt{25 - 16}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{9}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm 3}{2} \][/tex]
Thus, the solutions are:
[tex]\[ x = 4 \quad \text{and} \quad x = 1 \][/tex]
5. Relate [tex]\( x \)[/tex] back to [tex]\( r^4 \)[/tex]:
Recall that [tex]\( x = r^4 \)[/tex].
[tex]\[ r^4 = 4 \implies r = \pm \sqrt[4]{4} = \pm \sqrt{2} \][/tex]
[tex]\[ r^4 = 1 \implies r = \pm \sqrt[4]{1} = \pm 1 \][/tex]
6. Verify the valid values of [tex]\( r \)[/tex]:
Substitute [tex]\( r = \pm 1 \)[/tex] back into the original condition [tex]\( 1 - r^8 = 5 - 5r^4 \)[/tex]. We see that:
[tex]\[ 1 - 1^8 = 5 - 5 \cdot 1 \implies 0 \neq 0 \quad (\text{This does not hold true}) \][/tex]
Therefore, [tex]\( r = \pm 1 \)[/tex] are invalid solutions.
The only valid solutions for the common ratio [tex]\( r \)[/tex] are [tex]\( \sqrt{2} \)[/tex] and [tex]\( -\sqrt{2} \)[/tex]. Therefore, the correct answers are:
(a) [tex]\( \sqrt{2} \)[/tex]
(b) [tex]\( -\sqrt{2} \)[/tex]
Hence, the correct answer is:
(c) both
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.
