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Question

If the ninth term in the expansion of [log325x1+7+31/8log35x1+1]10is equal to 180 and x > 1 then show that x is equal to log5 15

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Solution

In (A+B)n,T9=nC8An8B8=180
We know that alogan=n

25x1+7=52(x1)+7

b=[(5x1+1)]1/8,n=10

T9=10C8A2B8=180

10C8(52(x1)+7)[(5(x1)+1)]1=180

Now put 5x1=t

10.961.2(t2+7)(t+7)1=180

t2+7=4(t+1)ort24t+3=0

or ( t - 3) (t - 1) = 0 t = 3 , 1

5x1=3,1or5x=15,5

x=log515 or x = 1

But x > 1 x=log515

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