Relations and Functions

Relations and Functions

Question

Simplify

space space space space space space space space sum from straight n space equals space 1 to 200 of straight i to the power of straight n

Answer
space space space space space space space space space space sum from 1 space equals space straight n to 200 of straight i to the power of straight n space equals space(i1 + i2 + i3 + i4  + i5  + i6 + i7 + i8 ) + ........+ i200 )

    = (i1 + i2 + i3 + i4  + i5  + i6 + i7 + i8 ) + ......+ (i197 + i198 + i199 + i200 )    
                                                                                                        ...(i)
Now, i4 = 1, putting in (i)
rightwards double arrow space space sum from 1 space equals space straight n to 200 of straight i to the power of straight n space equals space (i1 + i2 + i3 + 1) + (i4 . i + i4 .i2 +i4 . i3 +1) .... + (i196 i + i196 .i2 + i196 . i3 + 1)

= (i1 + i2 + i3 + 1) + (i1 + i2 + i3 + 1) + ...... + (i1 + i2 + i3 + 1)
= (i - 1 - i + 1) + (i - 1 - i + 1) + ...... (i - 1 - i + 1) +            ( i2 = -1,  i3 = 1)
= 0 + 0 + 0 ......+ 0 = 0

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