In question 10 part be , in the solution part what does p(n) mean please 9:581 MailAAA bartleby.com= bartlebyE Q&AMath / Advanced Math / Q&A Library / 10.) For all n 1, prov...10.) For all n1, prove the following by m...Get live help whenever you need Try bartlebytutor todayfrom online tutors!п(2n - 1)(2n + 1)for all n 1.(d) 1² + 3² + 5² + … … - + (2n – 1)² =3ding(e) 13 + 2³ + 3³ + ..+n³ =for all n 1.2 fr + 1, show that for any positive integer n,a(n+1 – 1)a + ar + ar“ +… . + ar" =r - 13. Use the Second Principle of Finite Induction to establish that for all n 1,a" – 1 = (a – 1)(a"-1 +a"-2 + a"-3+ ..+ a + 1)[Hint: a"+1 – 1 = (a + 1)(a" – 1) – a(a"-1 – 1).]4 Prove that the cube of any integer can be written as the difference of two squares. [Hint:Notice thattheiven² = (1³ + 2³ + . . + n³) – (1³ + 2³ + • … + (n – 1)').]5. (a) Find the values of n 7).(b) True or false? For positive integers m and n, (mn)! = m!n!and (m + n)! = m! + n!.6. Prove that n! n² for every integer n 4, whereas n! n³ for every integer n 6.T.Use mathematical induction to derive the following formula for all n 1:Drle1(1!) + 2(2!) + 3(3!) + · . · + n(n!) = (n+1)! – 18. (a) Verify that for all n 1,(2n)!2.6 10 14 ..... (4n – 2) =n!(b) Use part (a) to obtain the inequality 2" (n!)² 1.9. Establish the Bernoulli inequality: If 1 + a 0, then(1+ a)" 1+ nafor all n 1.10.) For all n 1, prove the following by mathematical induction:11 0.12. Consider the function defined by3n +1for n oddT(n) =for n evenconjecture is the claim that starting from any integer n 1, thT(n), T(T (n)), T(T(T (n)), .. , eventually reaches the integer 1 a.runs through the values 1 and 2. This has been verified for all n 1, prov...10.) For all n 1, prove the following by m...Get live help whenever you need Try bartlebyfrom online tutors!tutor todaySolution-Let p(n): +3+27= 2 - (n+2)Step 2For n: 1,L.H.S = 1R.H.S- 2- ( 1 +2) _ 2-ニ.. L'H.S = R.H.Sp(m) is taue for n= 1.ASsume thatP(m) is toue for n=K,PIK):늘 +2.32-(K+2)ス2223 +2K四Step 3レ

Question

In question 10 part be , in the solution part what does p(n) mean please 9:581 MailAAA bartleby.com= bartlebyE Q&AMath / Advanced Math / Q&A Library / 10.) For all n > 1, prov...10.) For all n>1, prove the following by m...Get live help whenever you need Try bartlebytutor todayfrom online tutors!п(2n - 1)(2n + 1)for all n > 1.(d) 1² + 3² + 5² + … … - + (2n – 1)² =3ding(e) 13 + 2³ + 3³ + ..+n³ =for all n > 1.2 fr + 1, show that for any positive integer n,a(n+1 – 1)a + ar + ar“ +… . + ar" =r - 13. Use the Second Principle of Finite Induction to establish that for all n > 1,a" – 1 = (a – 1)(a"-1 +a"-2 + a"-3+ ..+ a + 1)[Hint: a"+1 – 1 = (a + 1)(a" – 1) – a(a"-1 – 1).]4 Prove that the cube of any integer can be written as the difference of two squares. [Hint:Notice thattheiven² = (1³ + 2³ + . . + n³) – (1³ + 2³ + • … + (n – 1)').]5. (a) Find the values of n < 7 for which n! +1 is a perfect square (it is unknown whethern! +1 is a square for any n > 7).(b) True or false? For positive integers m and n, (mn)! = m!n!and (m + n)! = m! + n!.6. Prove that n! > n² for every integer n > 4, whereas n! > n³ for every integer n > 6.T.Use mathematical induction to derive the following formula for all n > 1:Drle1(1!) + 2(2!) + 3(3!) + · . · + n(n!) = (n+1)! – 18. (a) Verify that for all n > 1,(2n)!2.6 10 14 ..... (4n – 2) =n!(b) Use part (a) to obtain the inequality 2" (n!)² < (2n)! for all n > 1.9. Establish the Bernoulli inequality: If 1 + a > 0, then(1+ a)" > 1+ nafor all n > 1.10.) For all n > 1, prove the following by mathematical induction:11<2(a)122232n2n13n+2...+= 222 232n2n11. Show that the expression (2n)!/2"n! is an integer for all n > 0.12. Consider the function defined by3n +1for n oddT(n) =for n evenconjecture is the claim that starting from any integer n > 1, thT(n), T(T (n)), T(T(T (n)), .. , eventually reaches the integer 1 a.runs through the values 1 and 2. This has been verified for all n < 1016. CoOl 9:581 MailAAA bartleby.com= bartlebyE Q&AMath / Advanced Math / Q&A Library / 10.) For all n > 1, prov...10.) For all n> 1, prove the following by m...Get live help whenever you need Try bartlebyfrom online tutors!tutor todaySolution-Let p(n): +3+27= 2 - (n+2)Step 2For n: 1,L.H.S = 1R.H.S- 2- ( 1 +2) _ 2-ニ.. L'H.S = R.H.Sp(m) is taue for n= 1.ASsume thatP(m) is toue for n=K,PIK):늘 +2.32-(K+2)ス2223 +2K四Step 3レ

Accepted Answer

10(b) here given a series 
12+222+322+.....+n2n=2-n+22n ,   ∀n≥1
we have to prove this by using…

Let p(n): 23 27 = 2 - (n+2)