First slide

Binomial theorem for positive integral Index

Question

For each $n \in N, x^{2 n - 1} + y^{2 n - 1}$ is divisible by

Moderate

A

$x + y$
B

$(x + y)^{2}$
C

$x^{3} + y^{3}$
D

None of these

Solution

$\begin{array}{l} x^{2 n - 1} + y^{2 n - 1} \\ = (x + y) (x^{2 n - 2} - x^{2 n - 3} y + x^{2 n - 4} y^{2} - x^{2 n - 5} y^{3} + . . . . . \dots + y^{2 n - 2}) \end{array}$
Thus, $x^{2 n - 1} + y^{2 n - 1}$ 1 is divisible by x + y for all $n \in N$

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