Question

The value of expression 1.(2-w)(2-w2)+ 2.( 3-w )(3-w2)+ ..... +(n-1)(n-w)(n-w2), where w is an imaginary cube root of unity is .....

Answer 1

Answer:

Step-by-step explanation:

The expression

1(2-ω)(3-ω²) + 2(3-ω)(3-ω²) + ...........(n-1)(n-ω)(n-ω²)

can be written as

S = ∑$\left \{ {{n} \atop {k=2}} \right.$ [(k-1)(k-ω)(k-ω²)

= ∑[(k-1)(k² -k(ω+ω²)+ω³)]            (∑ from k=2 to k = n)

putting the value,

ω³ = 1

and  

ω + ω² = -1   in the equation we get,

S = ∑[(k-1)(k² +k+1)]

= ∑(k³-1)

=> S = (2³-1) + (3³-1) + ......(n³ -1)

=> S = (2³ + 3³ + ......n³)  -(n-1)

=> S = (1 + 2³ + 3³ + ......n³)  -(n)

=> S= [n(n+1)/2]² - n

which is the required expression.

Answer 2

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