The sum of first four terms of a geometric progression (G. P) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18} .$ If the product of first three terms of the G.P is 1,and the third term is a, then 2a is………………

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alok kumar singh | Contributor-Level 10

a month ago

Let a, ar, ar², ar³ are in G.P.

a + ar + ar² + ar³ = $\frac{65}{12}$

$\frac{1}{a} + \frac{1}{a r} + \frac{1}{a r^{2}} + \frac{1}{a r^{3}} = \frac{65}{18}$

${(a r)}^{2} r = \frac{3}{2} \Rightarrow a = \frac{2}{3}, r = \frac{3}{2}$

$∴$ third term = ar² = $\frac{2}{3} \times \frac{9}{4} = \frac{3}{2}$

$∴ α = \frac{3}{2} \Rightarrow 2 α = 3$

Let a, ar, ar2, ar3 are in G.P.a + ar + ar2 + ar3 = <math> <mrow> <mfrac> <mrow> <mn>6</mn> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816609755"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>a</mi> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>a</mi> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>8</mn> </mrow> </mfrac> </mrow> </math>      <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816609756"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>⇒</mo> <mi>a</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>, </mo> <mi>r</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>          <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816609757"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mo>∴</mo> </mrow> </math>   third term = ar2 = <math> <mrow> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>9</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1029" DrawAspect="Content" ObjectID="_1816609759"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mo>∴</mo> <mi>α</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>⇒</mo> <mn>2</mn> <mi>α</mi> <mo>=</mo> <mn>3</mn> </mrow> </math>          <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1030" DrawAspect="Content" ObjectID="_1816609760"> </o:OLEObject></xml><! [endif]->

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The sum of first four terms of a geometric progression (G. P) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18} .$ If the product of first three terms of the G.P is 1,and the third term is a, then 2a is………………

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The sum of first four terms of a geometric progression (G. P) is 6 5 1 2 and the sum of their respective reciprocals is 6 5 1 8 . If the product of first three terms of the G.P is 1,and the third term is a, then 2a is………………

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The sum of first four terms of a geometric progression (G. P) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18} .$ If the product of first three terms of the G.P is 1,and the third term is a, then 2a is………………