Z scores are reported as the same measurement unit as the raw scores.

19. \( 1+2+3+4+5+\ldots+n=\frac{n(n+1)}{2} \) (a) i) Show that this formula is true for the sum of the first 8 natural numbers. ii) Find the sum of the first 400 natural numbers. (b) i) Show that \( 2+4+6+8+\cdots+2 n=n(n+1) \). ii) Find the sum of the first 200 even numbers. iii) Find the sum of the first 200 odd numbers. (c) i) Use the formula at the beginning of the question to find the sum of the first \( 2 n \) natural numbers. ii) Find a formula, in its simplest form, for \( 1+3+5+7+9+\cdots+(2 n-1) \).

(iii) \( \tan \theta=\frac{3}{4}, \theta \) in quadrant III

\( \frac { d y } { d t } = ( a - y ) ( b - y ) \)

A ball of mass 0.8 kg flies through the air at low speed, so that air resistance is negligible. What is the net force acting on the ball while it is in motion? \( \vec{F}_{\text {net }}=0.8 \mathrm{~kg} \cdot 9.81 \frac{\mathrm{m}}{\mathrm{s}} 2 \) Which components of the ball's momentum will be changed by this force? y only What happens to the \( x \) component of the ball's momentum during its flight? It doesn't change. \( \hat{v} \) What happens to the \( y \) component of the ball's momentum during its flight? It decreases. \( \hat{v} x \) What happens to the \( z \) component of the ball's momentum during its flight? It doesn't change. \( \hat{v} \) In this situation, why is it legitimate to use the formula for average \( y \) component of velocity, \( v_{\text {avg, }}=\frac{v_{y i}+v_{y f}}{2} \), to update the \( y \) component of position? Check all that apply. The ball's velocity changes at a constant rate because the net force on the ball is constant. This formula for average velocity is always valid. The ball's speed is small compared to the speed of light.

Se dispone de un pentágono regular con apotema \( a=8.7 \), inscrito en una circunferencia de radio \( r=a+0.75 \mathrm{~cm} \). Sobre uno de sus lados se construye un triángulo equilatero, como se ve en la figura. ¿Cuál es la altura \( h \) de ese tríangulo equilaterop Justifique su respuesta en su evidencia.