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Question

If, A1=n+1n(min{|xn|,|x(n+1)|})dx,
A2=n+2n+1(|xn||x(n+1)|)dxA3=n+3n+2(|x(n+4)||x(n+3)|)dx and g(x)=A1+A2+A3, where nN, then

A
A1+A2+A3=9
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B
A1+A2+A3=94
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C
100n=1g(x)=9004
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D
100n=1g(x)=300
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Solution

The correct option is C 100n=1g(x)=9004
Here, min {|xn|,|x(n+1)|} can be shown as
A1=n+1n(min{|xn|,|x(n+1)|})dx=12×1×12=14 Now, A2=n+2n+1(|xn||x(n+1)|)dx=21(|t||t1|)dt Put x=n+tdx=dt=21(t(t1))dt=211dt=(t)21=1 and A3=n+3n+2(|x(n+4)||x(n+3)|)dx=32(|t4||t3|)dtPut x=n+tdx=dt=32((4t)(3t))dt=321dt=1 Also, g(x)=A1+A2+A3=14+1+1=94100n=1g(x)=g(1)+g(2)+g(3)++g(100)=94+94++94=9004 ​​​​​​

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