2. Domonstre por indução matemática que, para todo intelro positivo \( n \), é válida a proprosição \( P(n) \) : \( \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}} \)

2. (2.5 pts) Demuestra que para todo número natural \( n \), \[ 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1} \] ¿A partir de qué número \( n \) se empieza a cumplir la desigualdad estricta \( > \), sin el igual?

2) \( f(x)=\frac{x^{2}+x}{3 x^{2}-12} \)

A wooden artifact from an ancient tomb contains 40 percent of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made? (The half-life of carbon-14 is 5730 years.) years

27 Multiple Choice 1 point Solve the problem. The population of an animal species in a certain area is modeled by \( \mathbf{F}(\mathrm{t})=400 \mathrm{log}(2 t+3) \) where \( t \) is the time in months since the species was introduced to the area. Find the population of this species in the area 6 months after the species is introduced. 74 704 240

Find the equation for the exponential function that passes through the points \( (2,4) \) and \( (4,11) \).