My Generous Uncle

Calculus Level 2

My uncle says he'll deposit some money into my bank account. He'll deposit x x (where x < 2 x < 2 ) dollars on the first day, x 2 2 \dfrac{x^2}{2} dollars on the second day, x 3 4 \dfrac{x^3}{4} dollars on the third day, x 4 8 \dfrac{x^4}{8} dollars on the fourth day, and so on forever.

Which expression represents the amount of money that will be delivered into my bank account?

2 x 1 − x \frac{2x}{1 - x} x 1 − 2 x \frac{x}{1 - 2x} 2 x 2 − x \frac{2x}{2 - x} x 2 − x \frac{x}{2 - x}

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Andrew Ellinor by Andrew Ellinor , 33, USA

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1 solution

Daria Zafote
Jul 22, 2018

Use the formula for an infinite geometric sum: S ∞ = a 0 1 − r . S_\infty=\frac{a_0}{1-r}. Here, a 0 = x a_0=x and r = x 2 r=\frac{x}{2} . Substitute and multiply by 2 2 \frac{2}{2} : x ⋅ 2 ( 1 − x 2 ) ⋅ 2 = 2 x 2 − x \frac{x\cdot2}{(1-\frac{x}{2})\cdot2}=\frac{2x}{2-x}

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