Question

Prove that

$ln(1 + 3x + 2 {x}^{2} )$
$= 3x - \frac{5 {x}^{2} }{2} + \frac{9 {x}^{3} }{3} - \frac{17 {x}^{4} }{4} + ..............$


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Answer 1

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Answer 2

Solution:-

=> log ( 1 + 3x + 2x²)

=> Log (1+x) + log(1+2x)

=> [ x - x²/2 + x³/3 + ............ + (-1)^(n-1). xⁿ/n] + [ 2x - 4x²/2 + (2x)³/3 -.......]

=> 3x - 5x²/2 + 9x³/3 - 17x⁴/4 +............

Hence,

Coefficient of xⁿ is

=> (-1)^(n-1)/n + (-1)^(n+2n)/n

=> (-1)^(n-1) . (1+ 2ⁿ)

Hence Proved!!