Let Pn-1-1-321-1-62-1-2-nn-12- where n - 2n - N - If lim n - Pn-a-b- where a and b are coprime- then the valueof a - his

P_n= ∏_r=2^n(1 2r(r+1))^2 ⇒ P_n= ∏_r=2^n((r+2)(r 1)r(r+1))^2 ⇒ P_n= ∏_r=2^n(r+2r+1)^2 ∏_r=2^n(r 1r)^2 ⇒ P_n=(4^23^2×5^24^2×⋯×(n+2)^2(n+1)^2)(1^22^2×2^23^2×⋯×(n ...

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Byju's Answer

Standard XII

Mathematics

Indeterminate Forms

Let Pn=1-1/32...

Question

Let $P_{n} = {(1 - \frac{1}{3})}^{2} {(1 - \frac{1}{6})}^{2} \dots {(1 - \frac{2}{n (n + 1)})}^{2},$ where $n \geq 2 (n \in N) .$ If $lim n \to \infty P_{n} = \frac{a}{b},$ where $a$ and $b$ are coprime, then the value of $a + b$ is

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Solution

$P_{n} = n \prod r = 2 {(1 - \frac{2}{r (r + 1)})}^{2}$
$\Rightarrow P_{n} = n \prod r = 2 {(\frac{(r + 2) (r - 1)}{r (r + 1)})}^{2} \Rightarrow P_{n} = n \prod r = 2 {(\frac{r + 2}{r + 1})}^{2} n \prod r = 2 {(\frac{r - 1}{r})}^{2} \Rightarrow P_{n} = (\frac{4^{2}}{3^{2}} \times \frac{5^{2}}{4^{2}} \times \dots \times \frac{(n + 2)^{2}}{(n + 1)^{2}}) (\frac{1^{2}}{2^{2}} \times \frac{2^{2}}{3^{2}} \times \dots \times \frac{(n - 1)^{2}}{n^{2}}) \Rightarrow P_{n} = \frac{(n + 2)^{2}}{9} \times \frac{1}{n^{2}}$
$∴ lim n \to \infty P_{n} = \frac{1}{9} \equiv \frac{a}{b} \Rightarrow a + b = 10$

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Similar questions

Q. Let

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P_{n} = \frac{S_{2}}{S_{2} - 1} \cdot \frac{S_{3}}{S_{3} - 1} \cdot \frac{S_{4}}{S_{4} - 1} \cdot \cdot \cdot \frac{S_{n}}{S_{n} - 1}

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f (x) = ⎡ ⎢ ⎣ \begin{matrix} n & n + 1 & n + 2 {^{n} P}_{n} & {^{n + 1} P}_{n + 1} & {^{n + 2} P}_{n + 2} {^{n} C}_{n} & {^{n + 1} C}_{n + 1} & {^{n + 2} C}_{n + 2} \end{matrix} ⎤ ⎥ ⎦

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Let Pn=(1−13)2(1−16)2⋯(1−2n(n+1))2, where n≥2 (n∈N). If limn→∞Pn=ab, where a and b are coprime, then the value of a+b is

Pn=n∏r=2(1−2r(r+1))2 ⇒Pn=n∏r=2((r+2)(r−1)r(r+1))2⇒Pn=n∏r=2(r+2r+1)2 n∏r=2(r−1r)2⇒Pn=(4232×5242×⋯×(n+2)2(n+1)2)(1222×2232×⋯×(n−1)2n2)⇒Pn=(n+2)29×1n2 ∴limn→∞Pn=19≡ab⇒a+b=10

Let $P_{n} = {(1 - \frac{1}{3})}^{2} {(1 - \frac{1}{6})}^{2} \dots {(1 - \frac{2}{n (n + 1)})}^{2},$ where $n \geq 2 (n \in N) .$ If $lim n \to \infty P_{n} = \frac{a}{b},$ where $a$ and $b$ are coprime, then the value of $a + b$ is