Question

1. Use mathematical induction to show that, ∀n ≥ 3,
2n^{2} + 1 ≥ 5n

2. Letting s_{1} = 0, find a recursive formula for the
sequence 0, 1, 3, 7, 15,...

3. Evaluate. (a) 55mod 7. (b) −101 div 3.

4. Prove that the sum of two consecutive odd integers is divisible by 4

5. Show that if a|b then −a|b.

6. Prove or disprove: For any integers a,b, c, if a ∤ b and b ∤ c, then a ∤ c.

7. Use mathematical induction to show that, ∀n ≥ 0,
2|(3^{n} +1)

8. Solve the following. (a) List the first four terms of the
recursive sequence defined by s_{1} = 1 and ∀n ≥ 2,
s_{n} = (s_{n}−1 + 1) ^{2} . (b) Given that
∑_{i=1}^{n } (i= n(n+1)/2) , find the
sum 2 + 4 + 6 +...+ 200

Answer #1

Use mathematical induction to prove that for each integer n ≥ 4,
5n ≥ 2 2n+1 + 100.

Prove by mathematical induction that 5n + 3 is a
multiple of 4, or if it is not, show by induction that the
statement is false.

Use Mathematical Induction to prove that 3 | (n^3 + 2n) for all
integers n = 0, 1, 2, ....

Use Mathematical Induction to prove that for any odd integer n
>= 1, 4 divides 3n+1.

(a) use mathematical induction to show that 1 + 3 +.....+(2n +
1) = (n + 1)^2 for all n e N,n>1.(b) n<2^n for all n,n is
greater or equels to 1

Please note n's are superscripted.
(a) Use mathematical induction to prove that 2n+1 +
3n+1 ≤ 2 · 4n for all integers n ≥ 3.
(b) Let f(n) = 2n+1 + 3n+1 and g(n) =
4n. Using the inequality from part (a) prove that f(n) =
O(g(n)). You need to give a rigorous proof derived directly from
the definition of O-notation, without using any theorems from
class. (First, give a complete statement of the definition. Next,
show how f(n) =...

Use
mathematical induction to show that ?! ≥ 3? + 5? for all integers ?
≥ 7.

Use mathematical induction to prove that 3n ≥
n2n for n ≥ 0. (Note: dealing with the base case may
require some thought.
Please explain the inductive step in detail.

use mathematical induction to show that n> 2^n for all e
n,n>4

Prove by induction that 7 + 11 + 15 + … + (4n + 3) = ( n ) ( 2n
+ 5 )
Prove by induction that 1 + 5 + 25 + … + 5n-1 = ( 1/4 )( 5n – 1
)
Prove by strong induction that an = 3 an-1 + 5 an-2 is even with
a0 = 2 and a1 = 4.

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