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?

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

The exponential model \( A=177.2 e^{0.013 t} \) describe the population, \( A_{1} \) of a country in millions, \( t \) years after 2003 . Use the model to answer the following questions: a) what was the population of the country in 2003 ? b) By what percentage is the papulation of the country increasing each year?

Graph \( m(w)=-2 \cdot 2^{w} \)

Given: \( f(x)=2\left(x^{2}-1\right) \) Sketch function \( f \) on the diagram sheet. Clearly show all intercepts with the axes.