Pascals Triangle and Induction1.pdf

Pascal’s Triangle, Induction and the Binomial TheoremInduction:a. Suppose that the only currency were 3-Euro bills and 10-Euro notes. Showthat any amount greater than Euro 17 could be made from a combination ofthese notes.b. Prove that the following equality holds for every :c. Prove that for all .d. Show that for all .e. Prove that is divisible by 5 for all .f. Show that 21 divides for any positive integer n.g. Prove that is divisible by 4 for all .h. Let be the terms of a sequence defined by:and .Prove that each is divisible by 3.j. Prove that is a multiple of 6 for all natural numbers .Pascal’s Triangle:Using the Hockey Stick formula or in any other way, find a formula for:a)b)c)d)e)f) .g) ..Can you explain/prove the following patterns in Pascal’sTriangle?Pattern 1: Left-Right symmetry.Pattern 2: Hockey Stick Formulas (see diagram).Pattern 3: Sum of elements on a horizontal line.Hint: Use the Binomial formula for a special value of X.Pattern 4a: Sum of elements on a horizontal line, withalternate signs.Hint: Use the Binomial formula for a special value of X.Pattern 4b: Sum of elements on a horizontal line,skipping by twos.

Activity:In Pascal’s Triangle, colour all the odd numbers black, and leave the even numbers white. Doyou notice a pattern? Can you explain it?Look up the Sierpinski Triangle for more.Binomial formula exercises:Use the binomial expansion (or anything else you like) to solve these:a. Expand .b. If you expand what will the coefficient of be?c. If you expand what will the coefficient of be?d. If you expand is there a term independent of ? If so, what is it?e. What is the coefficient of in ?f. What is the coefficient of of in ?g. What is the highest power of inCan you find its coefficient?