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If n is a natural number and a multiple of 3 such that

#Mathematics #Algebra

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7 Answers

33

Abhishek Priyam ·2009-01-13 23:30:19

a_r=a_n-r

it helps or not [7]

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62

Lokesh Verma ·2009-01-13 23:45:35

isnt?

a_r=a_2n-2

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33

Abhishek Priyam ·2009-01-13 23:48:10

no a_r=a_2n-r

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33

Abhishek Priyam ·2009-01-13 23:48:34

[3]

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341

Hari Shankar ·2009-01-14 00:16:56

Consider (1-x³)ⁿ =(1+x+x²)ⁿ (1-x)ⁿ = Î£(-1)^r ⁿC_rx^3r

The coefficient of x^k on the right hand side is b_k = a_oⁿC_k - a₁ⁿC_k-1 + ....+(-1)ⁿa_kⁿC₀

If n is a multiple of 3, then let n = 3p

The coefficient of xⁿ on LHS is therefore (-1)^p ⁿC_p and on the RHS is

b_n = a_oⁿC_n - a₁ⁿC_n-1 + ....+(-1)ⁿa_nⁿC₀
which is equivalent to the given expression remembering that ⁿC_r = ⁿC_n-r

Hence the given expression equals (-1)^p ⁿC_p where p = n/3

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62

Lokesh Verma ·2009-01-17 09:37:57

phew.. good work prophet!!

Great work in fact...

Cant give 2 pinks for the same post.. or i wud have :)

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Ð”Ð²Ò¥Ñ—ÑuÏ now in medical c ·2009-01-20 23:06:02

thanxxxxxx....

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