Find the coefficient of 1/x in the expansion of (1+x)^n(1/x+x)^n
 I f the sum of binomial coefficients of the expansion (2x+1/x)/^n is equal to 256, then find the term independent of x

Asked by Nirbhay | 5th Nov, 2014, 05:54: AM

Expert Answer:

left parenthesis 1 plus x right parenthesis to the power of n T h e space left parenthesis r plus 1 right parenthesis t h space t e r m space f o r space t h e space a b o v e space e x p a n s i o n space i s comma t subscript r plus 1 end subscript equals to the power of n C subscript r space x to the power of r open parentheses x plus 1 over x close parentheses to the power of n T h e space left parenthesis r plus 1 right parenthesis t h space t e r m space f o r space t h e space a b o v e space e x p a n s i o n space i s t to the power of comma subscript r plus 1 end subscript equals to the power of n C subscript r. x to the power of n minus r end exponent. open parentheses 1 over x close parentheses to the power of r T h e space left parenthesis r plus 1 right parenthesis t h space t e r m space f o r space t h e space e x p a n s i o n comma left parenthesis 1 plus x right parenthesis to the power of n open parentheses x plus 1 over x close parentheses to the power of n space t subscript r plus 1 end subscript to the power of apostrophe apostrophe end exponent equals to the power of n C subscript r space x to the power of r. to the power of n C subscript r. x to the power of n minus r end exponent. open parentheses 1 over x close parentheses to the power of r equals open parentheses blank to the power of n C subscript r. to the power of n C subscript r close parentheses. x to the power of r. x to the power of n minus r end exponent. open parentheses 1 over x close parentheses to the power of r equals open parentheses blank to the power of n C subscript r close parentheses squared. x to the power of r plus n minus r minus r end exponent equals open parentheses blank to the power of n C subscript r close parentheses squared. x to the power of n minus r end exponent F o r space f i n d i n g space t h e space c o e f f i c i e n t space o f space 1 over x comma space space n minus r equals minus 1 rightwards double arrow r equals n plus 1 T h e space c o e f f i c i e n t space o f space open parentheses 1 over x close parentheses space i s space open parentheses blank to the power of n C subscript n plus 1 end subscript close parentheses squared comma space w h i c h space i s space n o t space p o s s i b l e comma space sin c e space n greater than r.  T h e space s u m space o f space t h e space b i n o m i a l space c o e f f i c i e n t s space o f space t h e space e x p a n s i o n space open parentheses 2 x plus 1 over x close parentheses to the power of n space i s to the power of n C subscript 0 plus to the power of n C subscript 1 plus to the power of n C subscript 2 plus to the power of n C subscript 3 plus............. plus to the power of n C subscript n equals 256 rightwards double arrow 2 to the power of n equals 256 rightwards double arrow n equals 8 T h e space left parenthesis r plus 1 right parenthesis t h space t e r m space o f space t h e space e x p a n s i o n space i s space open parentheses 2 x plus 1 over x close parentheses to the power of n t subscript r plus 1 end subscript equals to the power of n C subscript r left parenthesis 2 x right parenthesis to the power of n minus r end exponent. open parentheses 1 over x close parentheses to the power of r equals to the power of n C subscript r.2 to the power of n minus r end exponent. x to the power of n minus r minus r end exponent equals to the power of n C subscript r.2 to the power of n minus r end exponent. x to the power of n minus 2 r end exponent F o r space t e r m space i n d e p e n d e n t space o f space x space comma space w e space g e t comma n minus 2 r equals 0 rightwards double arrow r equals n over 2 equals 8 over 2 equals 4 therefore t subscript 4 plus 1 end subscript equals t subscript 5 equals to the power of 8 C subscript 4.2 to the power of 8 minus 4 end exponent equals to the power of 8 C subscript 4.2 to the power of 4 equals 70 cross times 16 equals 1120 T h e space t e r m space i n d e p e n d e n t space o f space x space i s space t h e space 5 t h space t e r m.

Answered by Prasenjit Paul | 7th Nov, 2014, 10:22: AM