Pascals Triangle and Induction1.pdf
Pascals Triangle and Induction1.pdf
Pascals Triangle and Induction1.pdf
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Pascal’s <strong>Triangle</strong>, Induction <strong>and</strong> the Binomial TheoremInduction:a. Suppose that the only currency were 3-Euro bills <strong>and</strong> 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:<strong>and</strong> .Prove that each is divisible by 3.j. Prove that is a multiple of 6 for all natural numbers .Pascal’s <strong>Triangle</strong>: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’s<strong>Triangle</strong>?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 <strong>Triangle</strong>, colour all the odd numbers black, <strong>and</strong> leave the even numbers white. Doyou notice a pattern? Can you explain it?Look up the Sierpinski <strong>Triangle</strong> for more.Binomial formula exercises:Use the binomial expansion (or anything else you like) to solve these:a. Exp<strong>and</strong> .b. If you exp<strong>and</strong> what will the coefficient of be?c. If you exp<strong>and</strong> what will the coefficient of be?d. If you exp<strong>and</strong> 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?