#17 W. Prove that for each natural number n, + ++.+ = 2- .Now consider 13+ 23 +33+... +k + (k+1) +11+1+D.+ (k + 1) .(k+1)(k+1+We know that we can write this ask (k+1)24(k+1)*4.Furthermore 4k1)+ (k+ 1) = 4Y(k+1) (k2+4k+4)(k+1) (+4(k+1))(k+1) (k+2)?(k+1)(k+1+1)Therefore BPMI. the statement is true for all natural numbers. .For problems 5 through 9, determine whether or not induction is an option asuethod of proof for the given statement. Note: You don't have to see ifthe inductive argument actually works, just state whether or not the statementrafiects the type of question for which induction would be an option.5. If n is any integer, then 8 divides 5" +2.3n-1+1.6. If n is a positive integer, then 2"> n2.7. For every integer n 2 1, 1(1!) +2(2!) + ..+n(n!) = (n+ 1)! – 1.8. If n is an integer, then n² +n+1 is odd.9. If r is a real number, then vr2 >r.Now we ask that you try to construct some induction proofs on your own:30. Prove that for any integer n,n 2 1, n° + 5n +6 is divisible by 3.11. Prove that for any natural number n, 2+5+8+.. + (3n –1) =Dn(3n+1)212. Prove that for any natural number n, 5"- 1 is divisible by 4.B. Prove that for every natural number n, 2°+2'+...+2"2n+1-1.%3D4. Prove that for any natural number n, 7|(9"- 2").Prove that for each natural number n, 5 + t..+%3D4n+1Prove that for each natural number n, 21| (4"+l+52n-1).n+2ER, f 1. Prove that for any natural number n,+ ..+ "n+1

Name: #17 W. Prove that for each natural number n, + ++.+ = 2- .Now consider
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Description: #17 W. Prove that for each natural number n, + ++.+ = 2- .Now consider 13+ 23 +33+... +k + (k+1) +11+1+D.+ (k + 1) .(k+1)(k+1+We know that we can write this ask (k+1)24(k+1)*4.Furthermore 4k1)+ (k+ 1) = 4Y(k+1) (k2+4k+4)(k+1) (+4(k+1))(k+1) (k+2)?(k+