The number of non−zero terms in the expansion of (1+3(√2) x)^9 +(1−3(√2) x)^9 is


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Question Number 63861 by gunawan last updated on 10/Jul/19

$The number of non - zero terms in the$ $expansion of {(1 + 3 \sqrt{2} x)}^{9} + {(1 - 3 \sqrt{2} x)}^{9} is$
Commented by kaivan.ahmadi last updated on 10/Jul/19

$e v e n p o w e r s o f x (o d d t e r m s) h a v e n o n z e r o c o f f$ $s o$ $i n {(1 + 3 \sqrt{2} x)}^{9} t h e s u m o f o d d c o f f i s$ $s = 3 \sqrt{2} + (\begin{matrix} 9 \\ 2 \end{matrix}) {(3 \sqrt{2})}^{7} + (\begin{matrix} 9 \\ 4 \end{matrix}) {(3 \sqrt{2})}^{4} + (\begin{matrix} 9 \\ 6 \end{matrix}) {(3 \sqrt{2})}^{2} + (\begin{matrix} 9 \\ 8 \end{matrix}) (3 \sqrt{2})$ $a n d i n {(1 + 3 \sqrt{2} x)}^{9} + {(1 - 3 \sqrt{2} x)}^{9} t h e s u m$ $o f c o f f i s 2 s$
Commented by mathmax by abdo last updated on 10/Jul/19

$w e h a v e {(1 + 3 \sqrt{2} x)}^{9} + {(1 - 3 \sqrt{2})}^{9} = \sum_{k = 0}^{9} C_{9}^{k} {(3 \sqrt{2} x)}^{k}$ $+ \sum_{k = 0}^{9} C_{9}^{k} {(- 1)}^{k} {(3 \sqrt{2})}^{k} = \sum_{k = 0}^{9} C_{9}^{k} (1 + {(- 1)}^{k}) {(3 \sqrt{2})}^{k} x^{k}$ $= \sum_{p = 0}^{[\frac{9}{2}]} C_{9}^{2 p} 2 \times 2^{p} 3^{2 p} x^{2 p} = \sum_{p = 0}^{4} C_{9}^{2 p} 2^{p + 1} 9^{p} x^{2 p}$ $\sum_{p = 0}^{4} a_{p} x^{2 p} w i t h a_{p} = 2^{p + 1} 9^{p} C_{9}^{2 p} w e s e e t h a t a_{p} ⩾ 0 . . . .$
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