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

1 answer:

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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