1.

1 +(2)/(4)+(2.5)/(4.8)+ (2.5.8)/(4.8.12) + (2.5.8.11)/(4.8.12.16)+ …isequal to

Answer»

`4^(-2//3 ) `
`3sqrt (16) `
`3sqrt4 `
`4^(3//2 ) `

SOLUTION : `1 +(2)/(4)+(2.5)/(4.8)+(2.5.8)/(4.8.12)+ … `
`(1 + x )^n= 1+ (nx )/(1 ! )+ (n (n - 1 ))/(2 ! ) x ^ 2+(n(n - 1 ) (n - 2 )) /(3! ) x ^ 3+ … `
On comparingthe abovetwoexpansions with , we get,
`therefore(nx )/( 1! )= (2 )/(4)"" `...(1)
` (n(n - 1 ))/(2 ! ) x ^ 2= (2.5)/(4.8)"" `...(2)
` (n(n - 1 )(n - 2 ))/(3 ! ) x ^ 3= (2.5.8)/(4.8.12) "" `...(3)
(2)` div`(1)
` ((n-1))/(2)x= (5)/(8)RARR(n-1)n =(5 )/(4) "" `...(4)
`(3)div(2) `
` ((n-2) x )/(3)=(8)/(12)rArr(n-2 ) x= 2 "" ...(5) `
` (4) -(5) `
`x =(5)/(4)- 2`
`x = (-3)/(4) `
`therefore((n-1))/(2)((-3)/(2))=(5)/(8) `
`thereforen =(-2)/(3) `
`therefore1 +(2)/(4)+(2.5)/(4.8)+ (2.5.8)/(4.8.12)+ ... =(1 - (3)/(4)) ^(-2/3) `
`= ((1)/(4)) ^(-2/3) `
`= 4 ^(2//3 ) `
`= 3sqrt(16)`


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