Find the first three terms in the expansion of - 1-x3-4 - -1-5x1-x2

Given eqn. can be written as y=(1 x)^ 2[(1+x)^34+(1+5x)^12] now expand each =[1+2x+3x^2. . . . .. ] [1+x.34 316.x^22!. . . . . .+1+5x.12 25x^22!.14... ...

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Byju's Answer

Standard XII

Mathematics

Numerically Greatest Term

Find the firs...

Question

Find the first three terms in the expansion of :
$\frac{(1 + x)^{\frac{3}{4}} + \sqrt{1 + 5 x}}{(1 - x)^{2}}$

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Solution

given eqn. can be written as

$y = (1 - x)^{- 2} [(1 + x)^{\frac{3}{4}} + (1 + 5 x)^{\frac{1}{2}}]$

now expand each
$= [1 + 2 x + 3 x^{2} . . . . . .] [1 + x . \frac{3}{4} - \frac{3}{16} . \frac{x^{2}}{2!} . . . . . . + 1 + 5 x . \frac{1}{2} - \frac{25 x^{2}}{2!} . \frac{1}{4} . . . . .]$

neglect powers higher than square

$= [1 + 2 x + 3 x^{2}] [2 + \frac{13 x}{4} - \frac{103 x^{2}}{32}]$

again neglect powers higher than square

$= [2 + \frac{29 x}{4} + \frac{297 x^{2}}{32}]$

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Find the first three terms in the expansion of :(1+x)34+√1+5x(1−x)2

given eqn. can be written as y=(1−x)−2[(1+x)34+(1+5x)12]now expand each =[1+2x+3x2......][1+x.34−316.x22!......+1+5x.12−25x22!.14.....]neglect powers higher than square=[1+2x+3x2][2+13x4−103x232] again neglect powers higher than square=[2+29x4+297x232]

Find the first three terms in the expansion of :
$\frac{(1 + x)^{\frac{3}{4}} + \sqrt{1 + 5 x}}{(1 - x)^{2}}$