1). If r is not equal to one, show that a + ar + ar^2+ ....+ ar^n= a( r^n+1 -1)/( r-1).2). Establish the Bernoulli inequality: If 1+a>0, then ( 1+a)^n >= 1+na.3). 1^2+ 3^2+ 5^2 +...+ ( 2n-1)^2 = n ( 2n-1)(2n+1)/3

Prove that if semicircles are constructed on each of the sides of a right triangle, then the area of the semicircle on the hypotenuse is equal to the sum of the areas of the semicircles on the two legs.

Prove by the indirect method: Given: MPN is not isosceles Prove: PMPN

Can you please show me how to prove this using the floor definition and let n = |_ x _| (floor of x). Please be explicit about it too!

Provide solution from the given problem on Pigeonhole Principle. Problem: In a room of 499 people, there exist two who share a birthday.

Which theorem proves that they are similar?

Find the clamped cubic spline that interpolates (0,0),(1,1),(2,2) and satisfies the condition: s’(0) = 1 and s’(2)  = 1.

Consider two points (x0, y0) and (x1, y1). Prove that the first order Lan￾grange polynonial is equivalent to linear interpolation.

(!) In the morning section of a calculus course, 2 of the 9 women and 2 of the 10 men receive the grade of A. In the afternoon section, 6 of the 9 women and 9 of the 14 men receive A. Verify that, in each section, a higher proportion of women than of men receive A, but that, in the combined course, a lower proportion of women than of men receive A. Explain!

Can u plz show work also?