Question-217895

Question Number 217895 by hardmath last updated on 23/Mar/25

Commented by mr W last updated on 23/Mar/25

question is wrong. C_n ^r is only defined for 0≤r≤n.

q u e s t i o n i s w r o n g . C_{n}^{r} i s o n l y d e f i n e d

f o r 0 ⩽ r ⩽ n .

Commented by hardmath last updated on 23/Mar/25

RHS must be 1/2^(n+1)

Commented by mr W last updated on 23/Mar/25

you didn′t answer my question. say n=10, what is meant with C_(10) ^(100) ?

y o u {d i d n}^{'} t a n s w e r m y q u e s t i o n .

s a y n = 10, w h a t i s m e a n t w i t h C_{10}^{100} ?

Commented by Frix last updated on 23/Mar/25

I think some people use C_k ^n but others C_n ^k ...

I think some people use C_{k}^{n} but others C_{n}^{k} \dots

Commented by mr W last updated on 23/Mar/25

for k=1 to ∞, no matter whether you use C_n ^k or you use C_k ^n , it′s the same problem. if your question is not clear, you can not expect a right solution.

f o r k = 1 t o \infty, n o m a t t e r w h e t h e r

y o u u s e C_{n}^{k} o r y o u u s e C_{k}^{n}, {i t}^{'} s t h e

s a m e p r o b l e m . i f y o u r q u e s t i o n i s

n o t c l e a r, y o u c a n n o t e x p e c t a r i g h t

s o l u t i o n .

Commented by hardmath last updated on 23/Mar/25

Using (-1)ᵏ = i²ᵏ and (-1)ᵏ⁺¹ = i²ᵏ⁺²:

[∑ₖ[1→+∞] (-1)ᵏ C(2k, n)]²
+ [∑ₖ[1→+∞] (-1)ᵏ⁺¹ C(2k-1, n)]²
= [∑ₖ[1→+∞] i²ᵏ C(2k, n)]²
+ i³.[∑ₖ[1→+∞] i²ᵏ⁻¹ C(2k-1, n)]²
= [Re{(1+i)ⁿ}]² + i⁴ [Im{(1+i)ⁿ}]²
= |(1+i)ⁿ|²
= |1+i|²ⁿ
= (√2)²ⁿ
= 2ⁿ

Answered by SdC355 last updated on 23/Mar/25

We don′t know what sequence {C_n ^k } is.....

We {don}^{'} t know what sequence {C_{n}^{k}} is \dots . .

Facebook Tweet Pin

Leave a Reply Cancel reply