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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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Answer:

Range = (-<em>∞, 5</em>)

Step-by-step explanation:

This is the absolute value function with transformation.

The parent function is f(x) = |x|

This function has a "negative" in front, so it makes it reflect about x axis

The -4 after x makes horizontal translation of 4 units right

the +5 at the end makes the function translate 5 units UP

<em>The graph is shown in the attached picture.</em>

<em>Looking at the graph, we can clearly see the range. The range is the allowed y-values. Hence, we can see that the </em><em>range is  -infinity to 5</em>

<em />

<em>answer is not properly given, so i can't choose from the options, but the answer is -∞, 5 to 5</em>

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4 years ago
Fill in the blank.
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O centroid

Step-by-step explanation:

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has three medians, one from each vertex, and they all intersect each other at the triangle's centroid.

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a(n)=4n

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n=19

The sum of any arithmetic sequence (series are infinite) is:

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The average of the first and last terms times the number of terms, in this case we found that n=19 so:

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We know the original price was $234, during the sale there was a 35% discount, let's found out what that is...

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100% + 35% = 135%

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152.1 * 1.35

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The price of the camera after the increase is $205.33, I hope this helps!

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