prove by mathematical induction if x<y then x<1/2(x+y)<y?

Question

it my calculus assignment i need help in solving it. please... i still have 2 more days before the deadline.

Answer 1

given x < y

not sure if this would be using mathmatical induction but this is how i would do it

take
x < 1/2 (x+y) < y
2x< x + y < 2y
x < y < 2y

x therefore x < 1/2 (x+y) < y

Answer 2

I would not use mathematical induction, but I would use these three axioms: 1) x < y implies x+z < y+z, 2) x < y and z > 0 implies xz < yz, and 3) commutative property: x+y = y+x.

First, I will prove x < 1/2(x+y).

x < y. Then, x+x < y+x. Hence, 2x < y+x. Then, 2x*1/2 < (y+x)*1/2. Hence, x < (y+x)*1/2. Then, x < 1/2(y+x). Hence, x < 1/2(x+y).

Now, I will prove 1/2(x+y) < y.

x < y. Then, x+y < y+y. Hence, x+y < 2y. Then, (x+y)*1/2 < 2y*1/2. Hence, (x+y)*1/2 < y. Then, 1/2(x+y) < y.

Since x < 1/2(x+y) and 1/2(x+y) < y, x < 1/2(x+y) < y. We are able to check this inequality to be correct by another axiom: transitive property: x < y and y < z implies x < z. x < 1/2(x+y) and 1/2(x+y) < y implies x < y, which is true.

Therefore, x < 1/2(x+y) < y. ∎

Answer 3

For induction, show your base case:

Induction is used for integer stuff.
has 3 steps:

x=1, y=2 show its true 1. base case

assume true for x = n, y=n+k, k>0

show true for x = n+1
just plug n+1 where x appears in the equation to be proved. Note also that this means that you take y to be n+1+k, then manipulate, algebraically, to get the equation back into the same form excepte that n+1 appears where n was, this establishes it's proof by induction.
That's the basic three step induction process.

Answer 4

given x
the problem is basically calling for you to show that
1/2(x+y) = 1/2(x) + 1/2(y)
which is true, because of the "distributive property"

so, if x then we can state with certainty that

x < 1/2(x) + 1/2(y) < y


take for example, if x=10 and y=20

10 < 1/2(10) + 1/2(20) < 20

simplified,

10 < 5 + 10 < 20
10 < 15 < 20

hope this helps

Answer 5

This is not an induction problem .... as induction solutions require a relationship that is sequential to prove.
This is straight algebra

x x < y so x + y < 2y (add y to both sides)

Thus 2x < x + y < 2y
Dividing everything by 2 gives
x < ½(x + y) < y