document.write( "Question 773490: 1/2+1/4+1/8+.....+1/2^n=1-1/2^n.\r
\n" ); document.write( "\n" ); document.write( "prove by mathematical induction that
\n" ); document.write( "above statement holds true for every
\n" ); document.write( "integer n belongs to N.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "HINT:to prove that 1-1/2^(k+1) \n" ); document.write( "

Algebra.Com's Answer #471648 by ramkikk66(644) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "To prove that:\r\n" );
document.write( "\r\n" );
document.write( "To prove it using induction:\r\n" );
document.write( "1) Confirm it is true for n = 1\r\n" );
document.write( "It is true since 1/2 = 1/2^1\r\n" );
document.write( "2) Assume it is true for some value of n = k\r\n" );
document.write( "i.e. \r\n" );
document.write( " ----> eqn (1)\r\n" );
document.write( "3) Now prove it is true for n = k+1\r\n" );
document.write( "i.e. the sum up to (k+1) terms = 1 - 1/2^(k+1)\r\n" );
document.write( "Proof:\r\n" );
document.write( "For n = k+1, the expression of the sum is:\r\n" );
document.write( "\r\n" );
document.write( "=\r\n" );
document.write( " ---> from eqn(1)\r\n" );
document.write( "=\r\n" );
document.write( " ---> taking common denominator 2^(k+1)\r\n" );
document.write( "=\r\n" );
document.write( " ---> simplifying the numerator\r\n" );
document.write( "=\r\n" );
document.write( "\r\n" );
document.write( "Proved!\r\n" );
document.write( "4) So we have proved that if the formula is true for n=k, it is true for \r\n" );
document.write( "n=k+1. Since it is true for n=1, it is proved by mathematical induction, that\r\n" );
document.write( "it is true for all n.\r\n" );
document.write( "Hope you got it :)\r\n" );
document.write( "

\n" ); document.write( " \n" ); document.write( "

\n" );