The sum of the last eight coefficients in the expansion of (1+x)^(16) is 2^(15) .


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Question Number 55788 by gunawan last updated on 04/Mar/19

$The sum of the last eight coefficients in$ $the expansion of {(1 + x)}^{16} is 2^{15} .$
Commented by gunawan last updated on 04/Mar/19

$True or false$
Commented by maxmathsup by imad last updated on 04/Mar/19

$w e h a v e {(x + 1)}^{16} = \sum_{k = 0}^{16} C_{16}^{k} x^{k} = \sum_{k = 0}^{16} a_{k} x^{k} \Rightarrow$ $a_{0} + a_{1} + . . . . + a_{7} = \sum_{k = 0}^{7} C_{16}^{k} b u t i f p (x) = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{n} x^{n} \Rightarrow$ $a_{o} + a_{1} + . . . . + a_{n} = p (1) a n d a_{o} + a_{1} + . . . + a_{p} = p (1) - a_{p + 1} - a_{p + 2} - . . . - a_{n} i n t h i s c a s e$ $p (x) = {(x + 1)}^{16} \Rightarrow a_{0} + a_{1} + . . . + a_{7} = p (1) - a_{8} - a_{9} - . . . . - a_{16}$ $b u t a_{8} = C_{16}^{8} a_{9} = C_{16}^{9} = C_{16}^{7} = a_{7}, a_{10} = C_{16}^{10} = C_{16}^{6} = a_{6}, a_{16} = a_{0}$ $2 (a_{0} + a_{1} + . . . . + a_{7}) = p (1) - a_{8} = 2^{16} - C_{16}^{8} = 2^{16} - \frac{16!}{8! 8!}$ $= 2^{16} - \frac{16!}{{(8!)}^{2}}$
Commented by maxmathsup by imad last updated on 04/Mar/19

$\Rightarrow a_{0} + a_{1} + . . . . + a_{7} = 2^{15} - \frac{1}{2} \frac{16!}{{(8!)}^{2}}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19

	$t o t a l t e r m s = 16 + 1 = 17$ $s u m o f c o e f f u c i e n t s$ ${(1 + x)}^{16} = 1 + 16 c_{1} x + 16 c_{2} x^{2} + 16 c_{3} x^{3} + . . + 16 c_{16} x^{16}$ $2^{16} = 16 c_{0} + 16 c_{1} + . . . + 16 c_{8} + 16 c_{9} + 16 c_{10} + . . + 16 c_{16}$ $w e h a v e t i f i n d t h e v a l u e o f$ $16 c_{16} + 16 c_{15} + 16 c_{14} + 16 c_{13} + 16 c_{12} + 16 c_{11} + 16 c_{10} + 16 c_{9} = k$ $a g a i n$ $16 c_{0} + 16 c_{1} + 16 c_{2} + 16 c_{3} + 16 c_{4} + 16 c_{5} + 16 c_{6} + 16 c_{7} = k$ $[s i n c e {n c}_{r} = {n c}_{n - r}]$ $n o w$ $n i n t h t e r m i s 16 c_{8} x^{8} \to c o e e f i c i e n t = 16 c_{8}$ $s o 2 k + 16 c_{8} = 2^{16}$ $2 k = 2^{16} - \frac{16!}{8! 8!}$ $k = 2^{15} - \frac{1}{2} (\frac{16!}{8! 8!})$ $r e q u i r e d a n s i s 2^{15} - \frac{1}{2} (\frac{16!}{8! 8!})$ $p l s c h e c k . . .$
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