Give a combinatorial proof the following statement 1(n−0) + 2(n−1) + 3(n−2) + 4(n−3) +···+ (n−1)2 + (n−0)1 =(n+ 2/3) Hint: on this one, you might consider thinking about subsets of the set{1,2,3,4, . . . , n−1, n, n+1, n+2}; hint 2: in each possible subset, pick the middle number

Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: Exercise 3. Give a combinatoric argument to prove the identity 2 2n (3.1) j=0 Hint: A subset of n…

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Q: (x – 2y2)5 ·+ bx³y* + •…· | ||

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Q: 10. Use induction to show that 1x2 + 2x3 + + n(n+1) = " for all n € N.

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Q: 6. Use the binomial theorem to prove that 3 = C(n,0) + 2C(n,1)+2²C(n, 2) + +2"C(n, n)

A: By using the binomial thorem to prove that, 3n=Cn, 0+2Cn, 1+22Cn, 2+...+2nCn, n

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Q: P(n, r) C(n, r) None of these n! (n - r)! Or! ||

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Q: Remark: For positive integers n, m, we have • (n) n (m) (n) + (m) = = (lcm (n, m)) (gcd (n, m))

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Q: Let P(n) be the statement that 1^2+2^3+· · ·+n(n + 1) = n(n + 1)(n + 2)/3 for the positive integer n…

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Q: (16) Prove that 22n – 1 is divisible by 3 for all n € N.

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