Problem for JEE Advanced
Find the value of n if z = n ± − i satisfies the equation i z 2 = 1 + z 2 + z 2 3 + z 3 4 + . . .
The answer is 1.
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2 solutions
Apologies, but would you care to elaborate more in your answer please?
Use the fact that ( n + 2 ) − 2 ( n + 1 ) + ( n ) = 0 , which you can check easily.
− 2 i z 2 = 1 + z 2 + z 2 3 + z 3 4 + z 4 5 + ⋯
− 2 i z 2 = 1 − z 2 − z 2 4 − z 3 6 − z 4 8 − ⋯
− 2 i z 2 = 1 + z 2 + z 2 1 + z 3 2 + z 4 3 + ⋯
Adding down the columns, we get i ( z 2 − 2 z + 1 ) = 1 , or ( z − 1 ) 2 = − i ) , or z = 1 ± − i .
The benefit of this one is that you don't need to memorize or look up a formula (for the arithmetico-geometric series).
The expression forms a arithmetico-geometric progression. By using the formula of sum to infinite terms of AGP, we get ( z − 1 ) 2 = − i and hence n = 1 .