document.write( "Question 354514: Consider the function f(x)=3x-5 and let f^n=fofofof....of where the composition is done n times. Find h(x)=f^n(x) \n" ); document.write( "
Algebra.Com's Answer #253316 by robertb(5830)\"\" \"About 
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\"f%5En%28x%29\" = \"%283%5En%29x\"-5(\"1%2B3%2B3%5E2\"+...+\"3%5E%28n-1%29\").
\n" ); document.write( "This can be proved by induction:
\n" ); document.write( "If \"n=1\", then \"f%5E1%28x%29=f%28x%29=3x-5%2A3%5E0\".
\n" ); document.write( "If \"n=2\", then \"f%5E2%28x%29=f%28x%29=3%283x-5%29x-5+=+%283%5E2%29x-3%2A5-5=%283%5E2%29x-5%2A%281%2B3%5E1%29\".
\n" ); document.write( "Assume for the purpose of induction that it is true for \"n+=+k\". have to show that it is also true for \"n=k\".
\n" ); document.write( "Assume this is true:
\n" ); document.write( "\"f%5E%28k-1%29%28x%29\" = \"%283%5E%28k-1%29%29x\"-5(\"1%2B3%2B3%5E2\"+...+\"3%5E%28k-2%29\").
\n" ); document.write( "Then \"f%5Ek%28x%29+=+f%5E%28k-1%29%28f%28x%29%29\",and
\n" ); document.write( "\"f%5Ek%28x%29=+f%5E%28k-1%29%283x-5%29\",
\n" ); document.write( "\"f%5Ek%28x%29+=3%5E%28k-1%29%283x-5%29\"-5(\"1%2B3%2B3%5E2\"+...+\"3%5E%28k-2%29\").
\n" ); document.write( "\"f%5Ek%28x%29+=%283%5Ek%29x-5%2A3%5E%28k-1%29\"-5(\"1%2B3%2B3%5E2\"+...+\"3%5E%28k-2%29\") by the distributive property.
\n" ); document.write( "\"f%5Ek%28x%29+=%283%5Ek%29x\"-5(\"1%2B3%2B3%5E2\"+...+\"3%5E%28k-2%29%2B3%5E%28k-1%29\") by associativity.
\n" ); document.write( "Therefore by induction the formula for \"f%5En+%28x%29\" is true for all positive integral n by induction. Now
\n" ); document.write( "\"f%5En%28x%29\" = \"%283%5En%29x\"-5(\"1%2B3%2B3%5E2\"+...+\"3%5E%28n-1%29\"). Since
\n" ); document.write( "\"1%2B3%2B3%5E2\"+...+\"3%5E%28n-1%29\"=\"%283%5En+-+1%29%2F2\",
\n" ); document.write( "then
\n" ); document.write( "\"f%5En%28x%29=%283%5En%29x-%285%2F2%29%283%5En+-+1%29\", or
\n" ); document.write( "\"f%5En%28x%29=%283%5En%29%28x-5%2F2%29%2B5%2F2%29\", \n" ); document.write( "
\n" );