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Given (1-2x+5x^2-10 x^3)(1+x)^n=1+a1x+a2...

Given `(1-2x+5x^2-10 x^3)(1+x)^n=1+a_1x+a_2x^2+` and that`a1 2=2a_2` then the value of `n` is.

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Verified by Experts

The correct Answer is:
6
`(1+2x+5x^(2)-10x^(3))[.^(n)C_(0) + .^(n)C_(1)x+.^(n)C_(2)x^(2)+ "…."]`
`= 1+a_(1)x+a_(2)x^(2) +"….."`
`rArr a_(1)= n-2` and `a_(2) = (n(n-1))/(2)-2n+5`
Given that `a_(1)^(2) = 2a_(2)`
`rArr (n-2)^(2) = n(n-1)-4n + 10`
or `n^(2) - 4n+4=n^(2)-5n+10`
or `n = 6`
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CENGAGE-BINOMIAL THEOREM-Exercise (Numerical)
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  3. If R is remainder when 6^(83)+8^(83) is divided by 49, then the value ...

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  4. The remainder, if 1+2+2^2++2^(1999) is divided by 5 is.

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  5. Given (1-2x+5x^2-10 x^3)(1+x)^n=1+a1x+a2x^2+ and thata1 2=2a2 then the...

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  6. Largest real value for x such that sum(k=0)^(4) ((3^(4-k))/((4-k)!))((...

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  7. The coefficient of x^(103) in (1+x+x^(2) +x^(3)+x^(4))^(199)(x-1)^(201...

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  8. The total number of different terms in the product (.^(101)C(0) - .^(1...

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  9. The constant term in the expansion of (log(x^(logx))-log(x^(2))100)^...

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  10. The value of sum(r=0)^(3) ""^(8)C(r)(""^(5)C(r+1)-""^(4)C(r)) is "".

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  11. The sum of the series (.^(101)C(1))/(.^(101)C(0)) + (2..^(101)C(2))/...

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  12. Let a=3^(1/(223))+1 and for all geq3,l e tf(n)=^n C0dota^(n-1)-^n C1do...

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  13. Let 1+ sum(r=1)^(10) (3^(r).""^(10)C(r)+r.""^(10)C(r)) = 2^(10)(alpha....

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  14. The value of lim(n rarroo) sum(r=0)^(n) (sum(r=0)^(r-1) (1)/(5^(n)).""...

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  15. If sum(r=0)^(n) ((r+2)/(r+1))""^(n)C(r)=(2^(8)-1)/(6).then n is equal ...

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  16. If S(n) = (.^(n)C(0))^(2) + (.^(n)C(1))^(2) + (.^(n)C(n))^(n), then ma...

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  17. The value of .^(40)C(0) xx .^(100)C(40) - .^(40)C(1) xx .^(99)C(40) + ...

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  18. The value of sum(0leiltjle5) sum(""^(5)C(j))(""^(j)C(i)) is equal to "...

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  19. If (1-x-x^(2))^(20) = sum(r=0)^(40)a(r).x^(r ), then value of a(1) + 3...

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  20. The value of 50 sum(r=1)^(49)(2r^(2) - 48r +1)/((50-r).""^(50)C(r)) is...

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