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Relation between Coefficient and Indices of x and y

if a0 , a...

Question

if $a_{0}, a_{1}, a_{2}$ ....be the coefficients in the expansion of (1 + x + x $^{2})^{n}$ in ascending powers f x , then prove that :
$a_{0} + a_{1} + a_{2} + . . . . + a_{n - 1} = \frac{1}{3} (3^{n} - a_{n})$

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Solution

Putting x = 1 in (1), we get
$a_{0} + a_{1} + a_{2} + . . . . + a_{r} + . . . . . + a_{2 n} = 3^{n}$ ......(2)
Since $a_{r} = a_{2 n - r}$ i.e. $a_{0} = a_{2} n, a_{1} = a_{2 n - 1}$ etc.
$2 (a_{0} + a_{1} + a_{2} + . . . . + a_{n - 1}) + a_{n} = 3^{n}$
$∴ a_{0} + a_{1} + a_{2} + . . . . + a_{n - 1} = \frac{1}{2} (3^{n} - a_{n})$

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