sec exercises prove by contradiction that the number of primes is infinite (ause induction to show that is always even (bgive direct proof in one or two sentences that is always even (cshow that is always divisible by three (dis aways divisible by explain your answer prove that is irrational explain why - ( ( ii= = = prove equation using mathematical induction prove equation using mathematical induction prove equation using mathematical induction find closed-form solution and prove (using inductionthat your solution is correct for the summation = prove that the sum of the first even numbers is (aindirectly by assuming that the sum of the first odd numbers is (bdirectly by mathematical induction give closed-form formula for the summation ni= where is an integer between and prove that fib( ) provefor > that ( ) = the following theorem is called the pigeonhole principle theorem when pigeons roost in holesthere must be some hole containing at least two pigeons (aprove the pigeonhole principle using proof by contradiction (bprove the pigeonhole principle using mathematical induction for this problemyou will consider arrangements of infinite lines in the plane such that three or more lines never intersect at single point (agive recurrence relation that expresses the number of regions formed by linesand explain why your recurrence is correct