Solve Q5, 6 explaining detailly each step MATHEMATICAL INDUCTION1. Prove by mathematical induction that for all positive integral values of n,(1+ 1)(2n + 1). Hence or otherwise, evaluate E"=2r²(r –1)Lr=13D212. Using mathematical induction, prove that 2"=1for all positive integral values of4r 2 - 12n+1n.3. Given thatn is a positive integer, prove by induction that, the 7"+ 5 is divisible by 6. Guessa common factor of 5"+ 3 for all positive integers n and prove your guess by induction.4. Prove by mathematical induction that for all positive integer n,2n(n+2)E=1 2r(r + 1) =5. (1) One of the statements that follow is true and the other is false. Prove the true statementand give a counter example to disprove the false statement.a. For all two dimensional vectors a, b, c, a.b = a.c →b = cb. For all positive real numbers a, b a+b Vab(ii)Prove, by mathematical induction that r(2r + 1) = -(n+1)(4n+5)6. a. Prove, by mathematical induction, that (4r + 3) = 2n2 + 5nb. Prove, by mathematical induction that /2 is not a rational number (you may assumethat the square of an odd integer is always odd).7. prove by mathematical induction that ,8. Prove by mathematical induction that 5"- 4n - 1 is divisible by 16, for all positive integersr (r+1)n+1n.9. Prove by mathematical induction that 8"-7n +6 is a multipie of 7, n e Z*n(n+1)(4n--1)10. Prove by induction that =1r(r - 1) :1211. Prove by use of mathematical induction, that , r12. DProve by mathematical induction that: 9"-1 is divisible by 8, for ail positive integers n. ii)(n + 1)23.Determine the value(s) of x for which oK+1/4.13. Prove by mathematical induction, that: r(r + 1)1.=n (n+1) (n+2), for all positive3.integers n.14. Prove by mathematical induction that: 2-1(2 + 3r) =-(3n + 7), for all positive integersn.Prove by mathematical induction that 3 + 7 is divisible by 8 for all integers n 0

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Solve Q5, 6 explaining detailly each step MATHEMATICAL INDUCTION1. Prove by mathematical induction that for all positive integral values of n,(1+ 1)(2n + 1). Hence or otherwise, evaluate E&#34;=2r²(r –1)Lr=13D212. Using mathematical induction, prove that 2&#34;=1for all positive integral values of4r 2 - 12n+1n.3. Given thatn is a positive integer, prove by induction that, the 7&#34;+ 5 is divisible by 6. Guessa common factor of 5&#34;+ 3 for all positive integers n and prove your guess by induction.4. Prove by mathematical induction that for all positive integer n,2n(n+2)E=1 2r(r + 1) =5. (1) One of the statements that follow is true and the other is false. Prove the true statementand give a counter example to disprove the false statement.a. For all two dimensional vectors a, b, c, a.b = a.c →b = cb. For all positive real numbers a, b a+b > Vab(ii)Prove, by mathematical induction that r(2r + 1) = -(n+1)(4n+5)6. a. Prove, by mathematical induction, that (4r + 3) = 2n2 + 5nb. Prove, by mathematical induction that /2 is not a rational number (you may assumethat the square of an odd integer is always odd).7. prove by mathematical induction that ,8. Prove by mathematical induction that 5&#34;- 4n - 1 is divisible by 16, for all positive integersr (r+1)n+1n.9. Prove by mathematical induction that 8&#34;-7n +6 is a multipie of 7, n e Z*n(n+1)(4n--1)10. Prove by induction that =1r(r - 1) :1211. Prove by use of mathematical induction, that , r12. DProve by mathematical induction that: 9&#34;-1 is divisible by 8, for ail positive integers n. ii)(n + 1)23.Determine the value(s) of x for which oK+1/4.13. Prove by mathematical induction, that: r(r + 1)1.=n (n+1) (n+2), for all positive3.integers n.14. Prove by mathematical induction that: 2-1(2 + 3r) =-(3n + 7), for all positive integersn.Prove by mathematical induction that 3 + 7 is divisible by 8 for all integers n > 0

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イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0