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The remainder, if 1+2+2^2+2^3+....+2^199...

The remainder, if `1+2+2^2+2^3+....+2^1999` in divided by 5 is

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The correct Answer is:
0
`1+2+2^(2)+2^(3)+"……"+2^(1999)`
`= (1(2^(2000)-1))/(2-1)`
`= 2^(2000) - 1`
`= (1-5)^(1000) - 1`
`= 1- .^(1000)C_(1).5+.^(1000)C_(2).5^(2)+"……"+.^(1000)C_(1000).5^(1000)-1`
which is divisible by 5.
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CENGAGE PUBLICATION-BINOMIAL THEOREM-Numerical
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  2. Let aa n db be the coefficients of x^3 in (1+x+2x^2+3x^3)^4a n d(1+x+2...

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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+2^3+....+2^1999 in divided by 5 is

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  5. Given (1-2x+5x^2-10x^3)(1+x)^n=1+a1x+a2x^2+.... and that a1^2=2a2, the...

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  6. Find the largest real value of 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 n geq3,l e tf(n)=^n C0dota^(n-1)-^n C1...

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  13. Let 1+sum(r=1)^(10)(3^r.^(10)Cr+r.^(10)Cr)=2^(10)(alpha. 4^5+beta) whe...

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  14. The value of lim(n->oo)sum(r=1)^(n)1/(5^n).^n Cr(sum(t=0)^(r-1).^r Ct ...

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  15. If underset(r=0)overset(n)sum((r+2)/(r+1)).^(n)C(r)=(2^(8)-1)/(6). the...

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

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  17. The value of .^(40)C0+.^40C1 + .^40C2 + + .^40C20

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

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