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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 | 21 Sep, 2016, 09:38: PM
answered-by-expert 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 | 22 Sep, 2016, 11:07: AM

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CBSE 11-commerce - Maths
<div>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.</div>
Asked by anoopshukla2902 | 21 Sep, 2016, 09:38: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
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