Final answer:
To find c, the given equation ∑ₘ=1[infinity]1/(1+c)ⁿ=9/2 represents an infinite geometric series. The sum S of this series is given by a/(1 - r), where a is the first term and r is the common ratio. By squaring both sides and solving the quadratic equation, the value of c is obtained.
Step-by-step explanation:
The student is asking to find the value of c that satisfies the equation of an infinite geometric series ∑ₘ=1[infinity]1/(1+c)ⁿ=9/2.
This is an infinite series with a common ratio r = 1/(1+c), where |r| < 1 for convergence. The sum of an infinite geometric series is given by S = a/(1 - r), where a is the first term of the series.
For the series given in the question, a = 1, and the sum S = 9/2. Therefore, 9/2 = 1/(1 - 1/(1+c)). This equation can be solved to find the value of c.
Start by solving for r: 9/2 = 1/(1 - r).
Transform the equation to 9/2 = (1+c)/(c).
Simplify and solve the resulting quadratic equation for c.