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Show that C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = 128

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Best answer

Since C0 + C1 + C2 + C3 + …… + Cn = 2n

Putting n = 8, we get

C0+ C1 + C2 + C3 + …… + C8 = 28

But, sum of even coefficients = sum of odd coefficients

∴ C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7

Let C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = k

Now, C0 + C1 + C2 + C3 + C4 + C5 + C6 + C7 + C8 = 256

∴ (C0 + C2 + C4 + C6 + C8 ) + (C1 + C3 + C5 + C7 ) = 256

∴ k + k = 256

∴ 2k = 256

∴ k = 128

∴ C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = 128

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