Maths-
General
Easy

Question

If vertical line x vertical line less than 1 fifth, the coefficient of x cubed in the expansion of  fraction numerator 1 over denominator left parenthesis 1 minus 5 x right parenthesis left parenthesis 1 minus 4 x right parenthesis end fraction is

  1. 369    
  2. 370    
  3. 371  
  4. 372  

The correct answer is: 369


    G i v e n comma
fraction numerator 1 over denominator open parentheses 1 minus 5 x close parentheses open parentheses 1 minus 4 x close parentheses end fraction
T o space f i n d comma
c o e f f i c i e n t space o f space x cubed

S o l u t i o n comma
fraction numerator 1 over denominator open parentheses 1 minus 5 x close parentheses open parentheses 1 minus 4 x close parentheses end fraction space equals space fraction numerator A over denominator open parentheses 1 minus 5 x close parentheses end fraction plus fraction numerator B over denominator open parentheses 1 minus 4 x close parentheses end fraction
fraction numerator 1 over denominator open parentheses 1 minus 5 x close parentheses open parentheses 1 minus 4 x close parentheses end fraction space equals space fraction numerator A open parentheses 1 minus 4 x close parentheses plus B open parentheses 1 minus 5 x close parentheses over denominator open parentheses 1 minus 5 x close parentheses open parentheses 1 minus 4 x close parentheses end fraction
1 equals A minus 4 A x plus B minus 5 B x

C o m p a r i n g space c o e f f i c i e n t s comma
A plus B equals 1
A equals 1 minus B space.... open parentheses 1 close parentheses
minus 4 A minus 5 B equals 0
4 A plus 5 B equals 0 space space.... open parentheses 2 close parentheses

S u b s t i t u t i n g space e q u a t i o n space 1 space i n space 2 comma
4 open parentheses 1 minus B close parentheses plus 5 B equals 0
4 minus 4 B plus 5 B equals 0
4 plus B equals 0
therefore B equals negative 4

S u b s t i t u t i n g space v a l u e space o f space B space i n space e q u a t i o n space 1 comma
A equals B minus 1 equals 1 minus open parentheses negative 4 close parentheses equals 5

N o w comma
fraction numerator A over denominator open parentheses 1 minus 5 x close parentheses end fraction plus fraction numerator B over denominator open parentheses 1 minus 4 x close parentheses end fraction equals fraction numerator 5 over denominator open parentheses 1 minus 5 x close parentheses end fraction plus fraction numerator negative 4 over denominator open parentheses 1 minus 4 x close parentheses end fraction equals 5 open parentheses 1 minus 5 x close parentheses to the power of negative 1 end exponent minus 4 open parentheses 1 minus 4 x close parentheses to the power of negative 1 end exponent

W e space k n o w space t h a t comma
open parentheses 1 minus x close parentheses to the power of negative 1 end exponent equals open parentheses 1 plus x plus x squared plus x cubed plus.... plus x to the power of n plus... close parentheses
F o r space x equals 4 x comma
open parentheses 1 minus 4 x close parentheses to the power of negative 1 end exponent equals open parentheses 1 plus 4 x plus open parentheses 4 x close parentheses squared plus open parentheses 4 x close parentheses cubed plus.... plus open parentheses 4 x close parentheses to the power of n plus... close parentheses
F o r space x equals 5 x comma
open parentheses 1 minus 5 x close parentheses to the power of negative 1 end exponent equals open parentheses 1 plus 5 x plus open parentheses 5 x close parentheses squared plus open parentheses 5 x close parentheses cubed plus.... plus open parentheses 5 x close parentheses to the power of n plus... close parentheses

W e space n e e d space t o space f i n d space t h e space c o e f f i c i e n t space o f space x cubed.
T h e r e f o r e comma
5 open parentheses 5 x close parentheses cubed minus 4 open parentheses 4 x close parentheses cubed equals 625 x cubed minus 256 x cubed equals 369 x cubed

H e n c e comma space t h e space c o e f f i c i e n t space o f space x cubed space i s space 369

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