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Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

If a=1+b+b2+b...

Question

If a = 1 + b + b² + b³ + ... to ∞, then write b in terms of a.

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Solution

$Here, a = 1, b, b^{2}, b^{3}, . . . \infty form an infinite G . P . ∴ S_{\infty} = a = 1 + b + b^{2} + b^{3} + . . . \infty = \frac{1}{1 - b} \Rightarrow a = \frac{1}{1 - b} \Rightarrow 1 - b = \frac{1}{a} \Rightarrow b = 1 - \frac{1}{a} ∴ b = \frac{a - 1}{a}$

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If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.

Here, a=1, b, b2, b3, ... ∞ form an infinite G.P. ∴ S∞=a =1+b+b2+b3+...∞ = 11-b⇒a = 11-b⇒1-b = 1a ⇒b = 1 - 1a∴ b = a-1a

If a = 1 + b + b² + b³ + ... to ∞, then write b in terms of a.

$Here, a = 1, b, b^{2}, b^{3}, . . . \infty form an infinite G . P . ∴ S_{\infty} = a = 1 + b + b^{2} + b^{3} + . . . \infty = \frac{1}{1 - b} \Rightarrow a = \frac{1}{1 - b} \Rightarrow 1 - b = \frac{1}{a} \Rightarrow b = 1 - \frac{1}{a} ∴ b = \frac{a - 1}{a}$