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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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Area=<br> Help me please!! Thanks :)
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Answer:

80π

Step-by-step explanation:

Circle O with radius IO:

large radius = IO = 12 = R

Circle O with diameter JK

IJ = JK = KL = 2 × IO = 24

IJ = JK = KL = 24/3 = 8

small radius = 8/2 = 4 = r

The shaded area is a semicircle of radius 12 minus a semicircle of radius 4 plus two semicircles of radius 4.

That is the same as

1 semicircle of radius 12 plus 1 semicircle of radius 4

A = πR²/2 + πr²/2

A = π(12² + 4²)/2

A = 160π/2

A = 80π

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A recipe requires 1/5 cup of milk for each 1/4 cup of water. How many cups of water are needed for each cup of milk?
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Answer:

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Step-by-step explanation:

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3 years ago
What’s the length of the mid segment of a trapezoid with bases of length 16 and 24
blagie [28]

Answer:

20

Step-by-step explanation:

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A large company wanted to know how the average salary of their employees had changed over the last year. the results of their st
Aleksandr-060686 [28]
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Which algebraic expression represents "the quotient of a number and thirty"? 30 + a 30a StartFraction 30 Over a EndFraction Star
Dima020 [189]

Answer:

the algebraic expression that represent the quotient of a number & thirty is 30 ÷ a

Step-by-step explanation:

The algebraic expression that represent the quotient of a number & thirty is shown below:

As we know that

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So here the quotient is 30

And the other number would be 30 ÷ a

Therefore the algebraic expression that represent the quotient of a number & thirty is 30 ÷ a

8 0
3 years ago
Read 2 more answers
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