Let A1, A2, A3,………. be squares such that for each <math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mrow> </math> the length of the side of An equals the length of diagonal of An+1. If the length of A1 is 12cm then smallest value of an for which area of An is less than one, is………………….

Let an be the side of square An a n = 2 a n + 1 <!- [endif]-><!- [if gte mso 9]><xml> </xml><! [endif]->a1 = 12an = 12 × ( 1 2 ) n − 1 <!- [endif]-><!- [if gte mso 9]><xml> </xml><! [endif]-> ( a n ) 2 < 1 <!- [endif]-><!- [if gte mso 9]><xml> </xml><! [endif]-> 1 4 4 2 ( n − 1 ) < 1 ⇒ 2 ( n − 1 ) > 1 4 4 ⇒ n − 1 ≥ 8 ⇒ n ≥ 9

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Sequences and Series Ncert Solutions Maths class 11th Maths Class 11th

Let A₁, A₂, A₃,………. be squares such that for each $n \geq 1,$ the length of the side of A_n equals the length of diagonal of A_n+1. If the length of A₁ is 12cm then smallest value of an for which area of A_n is less than one, is………………….

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Vishal Baghel | Contributor-Level 10

a month ago

Let a_n be the side of square A_n

$a_{n} = \sqrt[]{2} a_{n + 1}$

a₁ = 12

a_n = 12 $\times {(\frac{1}{\sqrt[]{2}})}^{n - 1}$

${(a_{n})}^{2} < 1$

$\frac{144}{2^{(n - 1)}} < 1$

$\Rightarrow 2^{(n - 1)} > 144$

$\Rightarrow n - 1 \geq 8$

$\Rightarrow n \geq 9$

Let an be the side of square An <math> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math> <!- [endif]-><!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1819630167"> </o:OLEObject></xml><! [endif]->a1 = 12an = 12 <math> <mrow> <mo>×</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> <!- [endif]-><!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1819630168"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>1</mn> </mrow> </math> <!- [endif]-><!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1819630169"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mfrac> <mrow> <mn>1</mn> <mn>4</mn> <mn>4</mn> </mrow> <mrow> <msup> <mrow> <mn>2</mn> </mrow> <mrow> <mrow> <mo> (</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </mfrac> <mo><</mo> <mn>1</mn> </mrow> </math> <math> <mrow> <mo>⇒</mo> <msup> <mrow> <mn>2</mn> </mrow> <mrow> <mrow> <mo> (</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>></mo> <mn>1</mn> <mn>4</mn> <mn>4</mn> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>≥</mo> <mn>8</mn> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mi>n</mi> <mo>≥</mo> <mn>9</mn> </mrow> </math>

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Let A₁, A₂, A₃,………. be squares such that for each $n \geq 1,$ the length of the side of A_n equals the length of diagonal of A_n+1. If the length of A₁ is 12cm then smallest value of an for which area of A_n is less than one, is………………….

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Let A1, A2, A3,………. be squares such that for each n ≥ 1 , the length of the side of An equals the length of diagonal of An+1. If the length of A1 is 12cm then smallest value of an for which area of An is less than one, is………………….

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Let A₁, A₂, A₃,………. be squares such that for each $n \geq 1,$ the length of the side of A_n equals the length of diagonal of A_n+1. If the length of A₁ is 12cm then smallest value of an for which area of A_n is less than one, is………………….