Algebra proofs

Several algebra tests are conducted using the evidence by recurrence. To demonstrate the power of recursion, we will prove an algebraic equation and a geometric formula with induction.

If you are not familiar with evidence by induction, carefully consider the evidence by recurrence given as a reference above. Otherwise, you may struggle with these tests in algebra below

Algebra equation:

Prove by recurrence that 1 + 2 + 4 + 8 +… + 2n-1 = 2n – 1

Step # 1:

Show that the equation is true for n = 2. n = 2 means adding the first two terms

1 + 2 = 3-22-1 = 4-1 = 3. Thus, it is true for n = 2

Just for fun, we will show it is also true for n = 4. n = 4 means adding the first 4 words

1 + 2 + 4 + 8 = 15-24-1 = 16-1 = 15. Thus, it is true also for n = 4

Step # 2:

Suppose that this is true for n = k

Just replace n by k

1 + 2 + 4 + 8 +… + 2 k-1 = 2 k – 1

Step # 3:

To prove this is true for n = k + 1

You need to write what it means for the equation be true for n = k + 1

Warning: It is true to write what it means, this is not the same that prove the equation. In fact, it shows you only what you need to prove

What this means for n = k + 1:

After replacing k by k + 1, you get:

1 + 2 + 4 + 8 +… + 2 k + 1-1 = 2 k + 1-1

1 + 2 + 4 + 8 +… + 2 k = 2 k + 1-1.

You can now complete the evidence using the hypothesis in step n ° 2 and then show that

1 + 2 + 4 + 8 +… + 2 k = 2 k + 1-1.

on the assumption, 1 + 2 + 4 + 8 +… + 2 k-1 = 2 k – 1

Ask yourself, “that the next term is seeking as”?

The last term now being 2 k-1, the next term should be 2 k + 1-1 = 2 k after replacing k by k + 1

Add 2 k to both sides of the hypothesis

1 + 2 + 4 + 8 +… + 2 k k-1 + 2 = 2 k – 1 + 2 k

The trick here is to see that 2 k + 2 k = 2 × 2 k = 21 × 2 k = 2 k + 1

1 + 2 + 4 + 8 +… + 2 k k-1 + 2 = 2 k – 1 + 2 k

= 2 × 2 k – 1

= 21 × 2 k – 1

= 2 k + 1-1

Geometric formula:

Show by recurrence that

the sum of the angles in a n-Gone = (n – 2) × 180 °

A couple of good observations before prove us it:

Comment # 1:

A n-Gone is a closed with n sides. For example,.

a n-gone with 4 sides is called a quadrilateral

a n-gone with 3 sides is called a triangle

Observation # 2:

Adding-sides-imageAfter a careful review of the above figure, you see that whenever you add a hand, you are also adding a triangle more

I am now ready to show the evidence.

Step # 1:

Show that the equation is true for n = 3. Note that n can be less than 3 because we cannot make a figure closed with only two sides or one side.

When n = 3, we get a triangle and the sum of the angles in a triangle is equal to 180 °

When n = 3, (3-2) × 180 ° = 1 × 180 ° = 180 °

When n = 4, you add a triangle more to obtain two triangles and the sum of the angles of two triangles is equal to 360 °

When n = 4, (4-2) × 180 ° = 2 × 180 ° = 360 °

Thus, the formula is true for n = 3 and n = 4

Step # 2:

Suppose that this is true for n = k

Just replace n by k

The sum of the angles in a k – gon = (k – 2) × 180 °

Step # 3:

To prove this is true for n = k + 1

You need to write what it means for the equation be true for n = k + 1

What this means for n = k + 1:

After replacing k by k + 1, you get:

The sum of the angles in a (k + 1) – gon = (k + 1-2) × 180 °

The sum of the angles in a (k + 1) – gon = (k – 1) × 180 °

You can now complete the evidence using the hypothesis in step n ° 2 and then show that

The sum of the angles in a (k + 1) – gon = (k – 1) × 180 °

on the assumption, the sum of the angles in a k – gon = (k – 2) × 180 °

Ask yourself, “that the next term is seeking as”?

Since the latter term is now k – gon or a silhouette alongside k, the next term should be a figure with k + 1 sides after replacing k by k + 1

Now, what we add to both sides?

First, recall the meaning of the addition of one side. This means that you will also add a triangle

(k + 1) gon = k – gon + (1 side or a triangle more)

(k + 1) = k – gon gon + a triangle more

(k + 1) gon = k – gon + 180 °

Add up to 180 ° to both sides of the hypothesis

The sum of the angles k – gon + 180 ° = (k – 2) × 180 ° + 180 °

The sum of the angles in a (k + 1) – gon = (k – 2) × 180 ° + 180 °

= 180 ° +-2 × 180 ° + 180 ° k

= 180 °-k + 1 × 180 °

= 180 °-(k + 1)

Other important algebra tests:

Following evidence of algebra not use recursion.

Proof of the Pythagorean theorem
A proof of the theorem of Pythagoras by President Garfield is clearly explained here

Proof of the quadratic formula
Here prove us the quadratic formula by completing the square

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