Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

Q: n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0

A:

Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

A:

Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

Q: 7n+1 20n+3 is irreducible for every natural number n. Prove that the fraction

A: Solution is given below:

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

A:

Q: Prove by induction that for positive integers n: E (312 + 2i + 5) = }(2n³ + 5n² + 13n)

A: To prove the given statement by mathematical induction.

Q: B. Prove that for all positive integers n, 2² +5² +8² +...+(3n − 1)² = n(6n² - 3n 2 1)

A:

Q: (c) Prove that 0 · Pn(0) +1· Pn(1) +2• Pn(2) +3· Pn(3)+...+n· Pn(n) = n! for all positive integers…

A: Let us consider all n! permutations of the set {1,2,3,..,n}. Given, pnk be the number of…

Q: Prove that k() = n(1) (n = 1) for integers n and k with 1 ≤ k ≤ n, using a: a) combinatorial proof;

A:

Q: Prove that 2n > n for all positive integers n.

A:

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

A:

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: prove that (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2n for all integers n greater than or…

A: We have to prove that  (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2 By mathematical…

Q: Prove the following formula for all n E N by induction: 1+ 2·2 + ... +n· 2"-l = 1 + (n – 1)2".

A:

Q: (b) Prove that for any positive integer n, Ej. 2 = (n – 1)2"+1 + 2 (c) n(n² – 1) Prove that for any…

A: b) Let the statement be P(n): 1·21+2·22+3·23+⋯ +n·2n(, or,∑j=1nj·2j )=(n-1)2n+1+2. For n=1, P(1) :…

Q: Q.No.6. Prove that by mathematically Induction 2n < (n+ 1)!, for all integers n 2.

A: (.)     2n &lt; (n+1)!      ∀integers  n≥2..................(*) (1)   now we check the above…

Q: 7

A:

Q: Prove that 2n>n² for all integers n25. Prove that 2n<n! for all integers n24.

A: Given below the detailed solution

Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

A:

Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

A:

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

A:

Q: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

A:

Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

Q: 7. Prove by induction. For any n2 0, E F = F,Fn+1• k=0

A:

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

Q: Use induction to prove: for any integer n > 1, (6j- 4) = 3n“ – n. j=1

A: We shall prove the given sum using Principle of Mathematical induction which means that a ) first we…

Q: Prove that n Σ 1 n²(n + 1)² i=1 Cor all positive integers n.

A:

Q: 1 (b) Prove by induction that (3i – 2)² = ;n(6n² – 3n – 1) for n 2 1

A: We have to prove ∑i=1n3i-22=12n6n2-3n-1 for n≥1 by using mathematical induction.

Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

A:

Q: Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

A:

Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: Show that Σ dn o(n) n for every positive integer n.

A:

Q: Prove by mathematical induction that for every positive integer n, II 1 1 1 %3D 3i – 1 3i 3i +1 (3n…

A: Mathematical induction

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

A:

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: (b) Prove by Induction that (3i – 1)(3i +2) = 3n +6n2 +n for n21 i=1

A:

Q: part ii and iii, thank you

A: According to pascal law,

Q: Prove that (f^2)_1 + (f^2)_2 + ⋯ + (f^2)_n = f_n*f_(n+1) when n is a positive integer.

A: We have to show that  f12+f22+f32+.....fn2=fnfn+1 where n is a positive integer and fn denotes the…

Q: Q.No.6. Prove that by mathematically Induction 2n < (n + 1)!, for all integers n 2 2.

A:

Q: 1. Prove by mathematical induction that for all positive integral values of n, 2=1 =- (n + 1)(2n +…

A: (1)-According to question given that∑r=1n r2    ....(1)Now  apply the sum formula we get that∑k=1n…

Q: Prove the following: for all integers n > 5, n2 < 2"

A:

Q: Q.No.6. Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

A:

Q: 1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the…

A:

Q: Prove by induction:  gn=gn-1+gn-2+gn-4   for n greater than or equal to 4, with g0=1, g1=1, g2=2,…

A:

Q: Prove that 1 n i(i+ 1) n+1 i=1 whenever n is a positive integer.

A:

Q: Prove that for all positive integers n, we have 6″ Σ (1) * _ "(-¹)" 2 k=1 k odd IG

A: