求等比数列公比r的可能值：已知第二项与第三项和为96、无穷项和为500

Let's work through this problem step by step using core geometric sequence properties.

Key Formulas to Start With #

First, recall these essential formulas for a geometric sequence with first term ( a_1 ) and common ratio ( r ):

• The ( n )-th term: ( a_n = a_1 r^{n-1} )
• The sum of an infinite geometric series (only valid when ( |r| < 1 )): ( S = \frac{a_1}{1 - r} )

Translate the Problem into Equations #

From the problem statement:

1. The sum of the second and third terms equals 96: ( a_2 + a_3 = 96 )
   Substitute the term formula to get: ( a_1 r + a_1 r^2 = 96 ), then factor out ( a_1 r ): ( a_1 r(1 + r) = 96 )
2. The infinite sum is 500: ( \frac{a_1}{1 - r} = 500 ), rearrange to solve for ( a_1 ): ( a_1 = 500(1 - r) )

Substitute and Simplify the Equation #

Plug ( a_1 = 500(1 - r) ) into the first equation:

500(1 - r) * r(1 + r) = 96

Simplify using ( (1 - r)(1 + r) = 1 - r^2 ):

500r(1 - r^2) = 96

Expand and rearrange into a standard cubic equation:

500r - 500r^3 = 96
500r^3 - 500r + 96 = 0

Divide all terms by 4 to simplify further:

125r^3 - 125r + 24 = 0

Factor the Cubic Equation #

Using the Rational Root Theorem, test ( r = \frac{1}{5} ) — it satisfies the equation:

125*(1/5)^3 - 125*(1/5) + 24 = 1 - 25 + 24 = 0

Factor out ( (5r - 1) ) (since ( r = 1/5 ) corresponds to ( 5r - 1 = 0 )):

125r^3 - 125r + 24 = (5r - 1)(25r^2 + 5r - 24)

Solve for ( r ) #

Now solve each factor:

1. ( 5r - 1 = 0 ) → ( r = \frac{1}{5} = 0.2 )
   Check the infinite sum condition: ( |0.2| < 1 ), which is valid.

2. Solve the quadratic ( 25r^2 + 5r - 24 = 0 ) using the quadratic formula ( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) (where ( a=25, b=5, c=-24 )):

r = \frac{-5 \pm \sqrt{5^2 - 4*25*(-24)}}{2*25}
r = \frac{-5 \pm \sqrt{25 + 2400}}{50}
r = \frac{-5 \pm 5\sqrt{97}}{50}
r = \frac{-1 \pm \sqrt{97}}{10}

Check the absolute value condition for each root:

• ( r = \frac{-1 + \sqrt{97}}{10} \approx 0.8849 ): ( |0.8849| < 1 ), valid.
• ( r = \frac{-1 - \sqrt{97}}{10} \approx -1.0849 ): ( |-1.0849| > 1 ), invalid (since the infinite sum only exists when ( |r| < 1 )).

Final Valid Values of ( r ) #

The two valid common ratios are:

• ( r = \frac{1}{5} )
• ( r = \frac{-1 + \sqrt{97}}{10} )

内容的提问来源于stack exchange，提问作者Fluellen