Question

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.

Answer #1

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

1. Use mathematical induction to show that, ∀n ≥ 3,
2n2 + 1 ≥ 5n
2. Letting s1 = 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...

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.

Prove by induction that 3^n ≥ 5n+10 for all n ≥ 3.
I get past the base case but confused on the inductive step.

Prove by mathematical induction: n3 – 7n + 3 is divisible by
3, for each integer n ≥ 1.

Prove by induction that 5n + 12n – 1 is divisible by 16 for all
positive integers n.

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

Prove by induction that n3-n is a multiple of 3.

Using mathematical induction (or otherwise) prove the following
statement: There is no m ∈ Z with 0 < m < 1.

3. Use Mathematical Induction on n to prove that if the TM
(above) is started with a blank tape, after 10 n + 4 steps the
machine will be in state 3 with the tape reading: . . . 0(0111)n01
↑ 1100 . . . .
That is, although there are three states with halting
instructions, show why none of those instructions is actually
encountered, and formulate this into a proof that this machine does
not halt when started with...

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