If y = 3 x + 6 x 2 + 10 x 3 + ……… , ∞ , then 3 1 y − 3 2 2 1.4 y 2 + 3 2 3 1.4.7 y 3 − …… ∞ is equal to

We have y = 3 x + 6 x 2 + 10 x 3 + …… ⇒ 1 + y = 1 + 3 x + 6 x 2 + 10 x 3 + … ⇒ 1 + y = ( 1 − x ) − 3 ⇒ 1 − x = ( 1 + y ) − 1/3 ⇒ x = 1 − ( 1 + y ) − 1/3 = 3 1 y − 3 2 ⋅ 2 1.4 y 2 + 3 2 ⋅ 3 1 ⋅ 4 ⋅ 7 y 3 − …

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MathematicsSequence & SeriesG.P.Medium2 minPYQ_2020

MathematicsMediumsingle choice

If $y = 3 x + 6 x^{2} + 10 x^{3} + \dots\dots\dots, \infty,$ then $\frac{1}{3} y - \frac{1.4}{3 ^{2} 2} y^{2} + \frac{1.4.7}{3 ^{2} 3} y^{3} - \dots\dots \infty$ is equal to

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Solution:

We have $y = 3 x + 6 x^{2} + 10 x^{3} + \dots\dots$ $\Rightarrow 1 + y = 1 + 3 x + 6 x^{2} + 10 x^{3} + \dots$ $\Rightarrow 1 + y = (1 - x)^{- 3} \Rightarrow 1 - x = (1 + y)^{- 1/3}$ $\Rightarrow x = 1 - (1 + y)^{- 1/3}$ $= \frac{1}{3} y - \frac{1.4}{3 ^{2} \cdot 2} y^{2} + \frac{1 \cdot 4 \cdot 7}{3 ^{2} \cdot 3} y^{3} - \dots$

Stream:BITSATSubject:MathematicsTopic:Sequence & SeriesSubtopic:G.P.

⏱ 2mℹ️ Source: PYQ_2020

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