Prove that for any natural n the following holds 1 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n +1

Q: Prove or disprove: There exists an integer a such that 14 | (3a — 5) and 21 | (2a + 3).

A:

Q: State the proof by Mathematical Induction of P(n), the statement that 1+3+5+ ...+(2n – 1)= n² for…

A:

Q: Prove by induction that 1 (1)4 + 1 1 + (4)7 (7)10 1 n (3n-2)(3n+1) 3n+1

A: Given that the statement 114+147+1710+...+13n-23n+1=n3n+1Let Sn=114+147+1710+...+13n-23n+1=n3n+1

Q: Prove that if n>1 is odd, then 1n+2n+3n+…+(n−1)n is divisible by n

A: by using mathematical induction, the given expression is proved and given below...

Q: 1.50 Prove: 2+4+6++ 2n n(n+1) 1.51 Prove: 1+4+7+.+ 3n – 2 = n(3n-1) 1.52 Prove: 12 +22+32 + ... +n²…

A:

Q: n(n+1)(2n+1) 6 In the proof that the sum of the first n squares is we showed P(0) and then P(k) ⇒…

A:

Q: Show that for any integer n, n² – 2 is not divisible by 3.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: b 2x 1! + 5x 2! + .+ (nt + 1)п! %3D п(п + 1)! + 10 x 3! + .. 2 3 + .. 1 n 1 + + + = 1 - 2! 3! 4! (п…

A: (b) Case 1 : n=1 2×1!+5×2!+10×3!+...+(12+1)1! = 1(1+1)!2×1! = 1(2)!2 = 2 Therefore, the result is…

Q: 1.54 Prove: 5 + 39 + .+ 4n-3)4n+1) = 4n+I п 1-5 5.9 9.13 (4n–3)(4n+1) | 1.55 Prove 7" – 2" is…

A:

Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

A:

Q: 7) A building is 300 feet tall. It casts a 165 foot long shadow. A statue near the building is 40…

A: Given: A building is 300 feet tall. It casts a 165 foot long shadow. A statue near the building is…

Q: Prove that for all positive integers n, 2 + 6 + 10 + ....+ (4n − 2) = 2n² -

A:

Q: Prove that for any natural n the following holds 1 1 1 + 5. 7 1 n + + (2n – 1)(2n + 1) 1.3 3.5 2n +…

A:

Q: Prove that the statements here are true for every positive integer n. 1/1 . 2 . 3 +1/2·3·4…

A: Given statement is, 11·2·3+12·3·4+13·4·5+.......+1n(n+1)(n+2)=n(n+3)4(n+1)(n+2)

Q: (2) Prove that there are no integersm and n such that 7m + 2ln - 15.

A: There are no integers m and n such that 7m+21n=15

Q: Prove that for all positive integers n, 13+23+33+…+n3=n(n+1)22.

A:

Q: 1. Prove by induction that for all n € N>1 we have (a) (b) (c) 2 +3 2.3 9+ (9 × 10) + (9 × 100) ++…

A: (a) Given 2n+3<2n·3 For n=1 L.H.S=21+3=5 R.H.S=21·3=6 Thus the inequality hold for n=1 Let…

Q: Dat

A:

Q: Prove that for any n E N, nº + 2n is divisible by 3.

A:

Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

A:

Q: (1) Prove that is a, b, and c are integers with c + 0 then ac = bc = a = b ( that is this a key step…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: (b) There exists an integer b such that for all integers n > b, n³ – 8 is prime.

A:

Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: (a) There exists two odd integers whose average is odd. 1 1 (b) There exists distinct positive…

A: (a) ans:- 1,5 (b) ans:- 2,3 &6 For detail...go through the solution

Q: By Induction. n Prove that į (3;²_zi) = 2n³ tn ²-n 2 Z=1 2 Goal: (2 (k+1) ³) + (k+₁)² – (k+1) 2 (1)…

A:

Q: Prove that for all n in J, 1 + 3 + 5 + • • • + (2n - 1) = n2

A: We have to prove 1+3+5+....+2n-1=n2

Q: Serrm f the Senes

A: ∑n=1∞1 / (n+4) (n+5) = 1/5*6 + 1/6*7+... =1/5 =0.2

Q: = There do not exist integers x and y such that 15x + 1 There is no integer n such that n² + 1 is…

A: Assume the result is true then obtain a contradiction.

Q: Prove that for all positive integers k < n, the equality n n n =2n-k k k i=k holds.

A:

Q: Prove that: n n (3) + (-²)=(-+ m m+1 n+1 m+1

A: We have to prove the given identity

Q: Eratosthene5-Legendre

A: It is problem of analytic number theory

Q: 1. Prove by induction that for all positive integers n, we have: 1+3+5+ + (2n 1) n2

A:

Q: How to prove these two questions?

A: As per the guidelines, we are entitled to solve only one question at a time. So, I am providing you…

Q: Use combinatorial proof to show that 10 (2n - 5 3 n - 2 n | %3D 10 k 10 – k

A: Given:

Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

A:

Q: (a) Show that for any integer n 21, 0 ≤ 22n² + 9n²+ 6n ≤ 37nª. Proof: Suppose n is any integer such…

A:

Q: Prove that for every integer n ≥ 0, 2 2 2 n n n 2n ()*² + ()*² ++ ()² = (2²). n n

A: