Operations on Binomial coefficients

Question

If $t_0, t_1, t_2, t_3,....., t_n$ are terms in the expansion of $(x+a)^n$ then prove that

$(t_0 - t_2 + t_4 -.....)^2 + (t_1 - t_3 + t_5 - .....)^2 = (x^2 + a^2)^n$

I have tried several ways to solve question including using expansion of conjugate ie $(x-a)^n$ and then trying addition annd subtraction of the series but i always reach a dead end. Plz suggest me an algebraic method to solve this question.....thanks in advance!!! Note- t0 indicates term no.

Answer 1

Hint. Let $i$ be the imaginary unit then the power $i^k$, for $k \geq 0$ is equal to $$1,i,-1,-i,1,i,-1,-i,1,i,-1,-i,\dots$$ Hence $$(x+ia)^n=(t_0 - t_2 + t_4 -\dots)+i(t_1 - t_3 + t_5 -\dots)$$ where $i$ is the imaginary unit.

Answer 2

HINT

Think about $(x+ia)^n$ where $i$ is the complex number, $i^2=-1$