The 3rd, 4th and 5th terms in expansion of (x+a)^n are respectively 84, 280 and 560. Find the values of x, a and n.

Asked by anoopshukla2902 | 21st Sep, 2016, 09:38: PM

Expert Answer:

straight T subscript 3 space end subscript equals space 84
straight T subscript 4 space end subscript equals space 280
straight T subscript 5 space end subscript equals space 560

straight T subscript straight r plus 1 end subscript space space equals space space to the power of straight n space straight C subscript straight r space space left parenthesis straight x right parenthesis to the power of straight n minus straight r end exponent left parenthesis straight a right parenthesis to the power of straight r

straight T subscript 3 space end subscript equals space to the power of straight n space straight C subscript 2 space space left parenthesis straight x right parenthesis to the power of straight n minus 2 end exponent left parenthesis straight a right parenthesis squared
84 equals fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 2 end exponent left parenthesis straight a right parenthesis squared................. left parenthesis straight i right parenthesis

straight T subscript 4 space end subscript equals space to the power of straight n space straight C subscript 3 space space left parenthesis straight x right parenthesis to the power of straight n minus 3 end exponent left parenthesis straight a right parenthesis cubed
280 equals fraction numerator straight n left parenthesis straight n minus 1 right parenthesis left parenthesis straight n minus 2 right parenthesis over denominator 6 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 3 end exponent left parenthesis straight a right parenthesis cubed................. left parenthesis ii right parenthesis

straight T subscript 5 space end subscript equals space to the power of straight n space straight C subscript 4 space space left parenthesis straight x right parenthesis to the power of straight n minus 4 end exponent left parenthesis straight a right parenthesis to the power of 4
560 equals fraction numerator straight n left parenthesis straight n minus 1 right parenthesis left parenthesis straight n minus 2 right parenthesis left parenthesis straight n minus 3 right parenthesis over denominator 24 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 4 end exponent left parenthesis straight a right parenthesis to the power of 4................. left parenthesis iii right parenthesis


To space elemenate space space straight x space and space straight a space terms space multiply space left parenthesis straight i right parenthesis space and space left parenthesis iii right parenthesis space and space divide space the space product space by space left parenthesis ii right parenthesis
fraction numerator 84 cross times 560 over denominator left parenthesis 280 right parenthesis squared end fraction equals fraction numerator open square brackets fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 2 end exponent left parenthesis straight a right parenthesis squared close square brackets open square brackets fraction numerator straight n left parenthesis straight n minus 1 right parenthesis left parenthesis straight n minus 2 right parenthesis left parenthesis straight n minus 3 right parenthesis over denominator 24 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 4 end exponent left parenthesis straight a right parenthesis to the power of 4 close square brackets over denominator open square brackets fraction numerator straight n left parenthesis straight n minus 1 right parenthesis left parenthesis straight n minus 2 right parenthesis over denominator 6 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 3 end exponent left parenthesis straight a right parenthesis cubed close square brackets squared end fraction
rightwards double arrow 3 over 5 equals fraction numerator 3 left parenthesis straight n minus 3 right parenthesis over denominator 4 left parenthesis straight n minus 2 right parenthesis end fraction
rightwards double arrow 5 straight n minus 15 equals 4 straight n minus 8
rightwards double arrow straight n space equals space 7........................ left parenthesis iv right parenthesis


Divide space equation space eq space left parenthesis ii right parenthesis space space by space eq space left parenthesis straight i right parenthesis

280 over 84 equals fraction numerator fraction numerator straight n left parenthesis straight n minus 1 right parenthesis left parenthesis straight n minus 2 right parenthesis over denominator 6 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 3 end exponent left parenthesis straight a right parenthesis cubed over denominator fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction space left parenthesis straight x right parenthesis to the power of straight n minus 2 end exponent left parenthesis straight a right parenthesis squared end fraction
rightwards double arrow fraction numerator left parenthesis straight n minus 2 right parenthesis over denominator 3 end fraction open parentheses straight a over straight x close parentheses equals 280 over 84
rightwards double arrow fraction numerator left parenthesis 7 minus 2 right parenthesis over denominator 3 end fraction open parentheses straight a over straight x close parentheses equals 280 over 84............... left square bracket from space left parenthesis iv right parenthesis right square bracket
rightwards double arrow straight x equals straight a over 2.................... left parenthesis straight v right parenthesis


Using space left parenthesis iv right parenthesis space and space left parenthesis straight v right parenthesis space in space left parenthesis straight i right parenthesis space we space get

84 equals fraction numerator 7 left parenthesis 7 minus 1 right parenthesis over denominator 2 end fraction space left parenthesis straight a over 2 right parenthesis to the power of 7 minus 2 end exponent left parenthesis straight a right parenthesis squared
84 equals 21 cross times straight a to the power of 7 over 2 to the power of 5
2 to the power of 7 equals straight a to the power of 7
straight a equals 2

from space left parenthesis straight v right parenthesis
straight x equals straight a over 2 equals 2 over 2 equals 1

So comma space straight x equals 1 comma space straight a space equals space 2 space and space straight n space equals space 7.

Answered by Vijaykumar Wani | 22nd Sep, 2016, 11:07: AM