document.write( "Question 1117569: if the coefficient of x^8, x^9 and x^10 in the expansion of (1+x)^n are in arithmetic progression, find the values of n where n is a positive integer \n" ); document.write( "
Algebra.Com's Answer #732686 by greenestamps(13200)![]() ![]() You can put this solution on YOUR website! \n" ); document.write( "The coefficients are.... \r\n" ); document.write( "C(n,8) = (n(n-1)...(n-6)(n-7))/8! \n" ); document.write( "We need to find the value(s) of n for which the three coefficients are in arithmetic progression -- i.e., for which C(n,9) is the arithmetic mean of C(n,8) and C(n,10). \n" ); document.write( "An interesting problem; but the algebra works out relatively easily.... \r\n" ); document.write( "(n(n-1)...(n-6)(n-7)(n-8))/9! = ((n(n-1)...(n-6)(n-7))/8!+(n(n-1)...(n-6)(n-7)(n-8)(n-9))/10!)/2 \n" ); document.write( "Multiply by 2*10! and cancel the common factors n through n-7: \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "The two solutions are n=14 and n=23. \n" ); document.write( "Check: \n" ); document.write( "For n=14, the coefficients are 1001, 2002, and 3003; 2002 = (1001+3003)/2. \n" ); document.write( "For n=23, the coefficients are 490314, 817190, and 1144066; 817190 = (490314+1144066)/2 \n" ); document.write( "DONE! \n" ); document.write( " |