The-sum-of-the-last-eight-coefficients-in-the-expansion-of-1-x-16-is-2-15-

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} + \dots . + 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} + \dots + a_{n} x^{n} \Rightarrow

a_{o} + a_{1} + \dots . + a_{n} = p (1) a n d a_{o} + a_{1} + \dots + a_{p} = p (1) - a_{p + 1} - a_{p + 2} - \dots - a_{n} i n t h i s c a s e

p (x) = {(x + 1)}^{16} \Rightarrow a_{0} + a_{1} + \dots + a_{7} = p (1) - a_{8} - a_{9} - \dots . - 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} + \dots . + 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} + \dots . + 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} + \dots + 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 \dots