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spin [16.1K]
3 years ago
6

Use mathematical induction to prove that the following statement is true for every positive integer n

Mathematics
1 answer:
Dafna1 [17]3 years ago
5 0
<h3>to prove </h3><h3> </h3><h3>8 </h3><h3>+ </h3><h3>16 </h3><h3>+ </h3><h3>24 </h3><h3>+ </h3><h3>... </h3><h3>+ </h3><h3>8 </h3><h3>n </h3><h3>= </h3><h3>4 </h3><h3>n </h3><h3>( </h3><h3>n </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>− </h3><h3>− </h3><h3>− </h3><h3>− </h3><h3>( </h3><h3>* </h3><h3>) </h3><h3> </h3><h3> let  </h3><h3>T </h3><h3>n </h3><h3>= </h3><h3>4 </h3><h3>n </h3><h3>( </h3><h3>n </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3> </h3><h3>(1) verify for  </h3><h3>n </h3><h3>= </h3><h3>1 </h3><h3> </h3><h3>L </h3><h3>H </h3><h3>S </h3><h3>= </h3><h3>8 </h3><h3> </h3><h3>R </h3><h3>H </h3><h3>S </h3><h3>= </h3><h3>4 </h3><h3>× </h3><h3>1 </h3><h3>( </h3><h3>1 </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>= </h3><h3>4 </h3><h3>× </h3><h3>2 </h3><h3>= </h3><h3>8 </h3><h3> </h3><h3>∴ </h3><h3>true for  </h3><h3>n </h3><h3>= </h3><h3>1 </h3><h3> </h3><h3># to show </h3><h3> </h3><h3>T </h3><h3>k </h3><h3>⇒ </h3><h3>T </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3> </h3><h3>assume true for  </h3><h3>T </h3><h3>k </h3><h3>= </h3><h3>4 </h3><h3>k </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3> </h3><h3>need to show </h3><h3> </h3><h3>T </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>= </h3><h3>4 </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>2 </h3><h3>) </h3><h3> </h3><h3>add next term to to both sides of  </h3><h3>( </h3><h3>* </h3><h3>) </h3><h3> </h3><h3>8 </h3><h3>+ </h3><h3>16 </h3><h3>+ </h3><h3>24 </h3><h3>+ </h3><h3>... </h3><h3>+ </h3><h3>8 </h3><h3>k </h3><h3>+ </h3><h3>8 </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>= </h3><h3>4 </h3><h3>k </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>+ </h3><h3>8 </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3> </h3><h3>∴ </h3><h3>T </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>= </h3><h3>4 </h3><h3>k </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>+ </h3><h3>8 </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3> </h3><h3>= </h3><h3>4 </h3><h3>( </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3>) </h3><h3>[ </h3><h3>k </h3><h3>+ </h3><h3>2 </h3><h3>] </h3><h3>= </h3><h3>T </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3> </h3><h3>i </h3><h3>. </h3><h3>e </h3><h3>. </h3><h3>T </h3><h3>k </h3><h3>⇒ </h3><h3>T </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3> as required </h3><h3> </h3><h3>#(3) conclusion </h3><h3> </h3><h3>statement true for  </h3><h3>T </h3><h3>1 </h3><h3> </h3><h3>∵ </h3><h3>T </h3><h3>k </h3><h3>⇒ </h3><h3>T </h3><h3>k </h3><h3>+ </h3><h3>1 </h3><h3> </h3><h3>T </h3><h3>1 </h3><h3>⇒ </h3><h3>T </h3><h3>2 </h3><h3>⇒ </h3><h3>T </h3><h3>3 </h3><h3>⇒ </h3><h3>... </h3><h3>∀ </h3><h3>n </h3><h3>∈ </h3><h3>N</h3>
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Three consecutive integers have a sum of 56. what are the three integers
denis23 [38]

Answer:

impossible

Step-by-step explanation:

X + X + 1 + X + 2 = 56. To solve for X, you first add the integers together and the X variables together. Then you subtract 3 from each side, followed by dividing by 3 on each side. ...

3X + 3 = 56. 3X + 3 - 3 = 56 - 3.

3X = 53. 3X/3 = 53/3.

X = 17 2/3. Since 17 2/3 is not an integer, there is no true answer to this problem.

8 0
3 years ago
Matt is showing his work in simplifying (6.2 − 1.6) − 4.4 + 7.8. Identify any error in his work or reasoning. Step 1: (6.2 − 1.6
Black_prince [1.1K]

Answer:

The correct option is (d)-He wrote commutative instead of associative in Step 2.

Step-by-step explanation:

Commutative property: a+b=b+a

Associative property: (a+b)+c=a+(b+c)

Now consider the provided expression.

(6.2 − 1.6) − 4.4 + 7.8

Step 1

(6.2 − 1.6) − 4.4 + 7.8

Step 2 Use Associative property

6.2 + (−1.6 − 4.4) + 7.8

Step 3: Simplify

6.2 − 6 + 7.8

Step 4: Use commutative property

14 − 6

Step 5: Simplify

14 − 6 = 8

Hence, the wrong step is step 2. He wrote commutative instead of associative in Step 2.

Therefore, the correct option is (d)-He wrote commutative instead of associative in Step 2.

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3 years ago
Consider the system: y = 3x + 5 y = ax + b What values for a and b make the system inconsistent? What values for a and b make th
AnnyKZ [126]
<span>When a = 3 and b ≠ 5, the system will be inconsistent because the lines will be parallel. When a = 3 and b = 5, the system will be consistent and dependent because they represent the same line.</span>
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roger bought 8 small bricks for his garden for $5.46. the total weight of the bricks was 6 1/4 pounds how many pounds does one b
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Step-by-step explanation:

the weight of 8 brick is 6.25 pound

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Alex_Xolod [135]

Answer:

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Step-by-step explanation:

I believe the answer is 12 boys. I hope this helps!

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