Let a -41 - 401-1 and for each n - 2- let b n - n C 1- n C 2- a - n C 3- a 2- n C n - a n -1- If the value of b 2006- b 2005 is 4 k-where k - N- then the value of k is

Given, a=4^1/401 1 ⇒ 1+a=4^1/401 b_n=^n nbsp;C_1+^n nbsp;C_2·a+^n nbsp;C_3 nbsp;· a^2+⋯ +^n nbsp;C_n nbsp;· a^n 1 =1a[^n nbsp;C_1 nbsp;· a+^n ...

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Binomial Expression

Let a =41 / 4...

Question

Let $a = 4^{1 / 401} - 1$ and for each $n \geq 2,$ let $b_{n} =^{n} C_{1} +^{n} C_{2} \cdot a +^{n} C_{3} \cdot a^{2} + \dots +^{n} C_{n} \cdot a^{n - 1} .$ If the value of $b_{2006} - b_{2005}$ is $4^{k},$ where $k \in N,$ then the value of $k$ is

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a = (4^{1 / 401} - 1)

n \geq 2,

b_{n} =^{n} C_{1} +^{n} C_{2} \cdot a +^{n} C_{3} \cdot a^{2} + \dots +^{n} C_{n} \cdot a^{n - 1} .

(b_{2006} - b_{2005})

4^{k}

k \in N,

k

a = (4^{1 / 401} - 1)

b_{n} =_{1}^{C} +^{n} C_{2} . a +^{n} C_{3} . a^{2} + . . . . . . . . . . . . +^{n} C_{n} . A^{n - 1}

(b_{2006} - b_{2005})

a = (4^{1 / 401} - 1)

n \geq 2,

b_{n} =^{n} C_{1} +^{n} C_{2} \cdot a +^{n} C_{3} \cdot a^{2} +^{n} C_{n} \cdot a^{n - 1} .

b_{2020} - b_{2019}

a = 4^{1 / 401} - 1

n \geq 2,

b_{n} =^{n} C_{1} +^{n} C_{2} \cdot a +^{n} C_{3} \cdot a^{2} + \dots +^{n} C_{n} \cdot a^{n - 1} .

b_{2006} - b_{2005}

4^{k},

k \in N,

k

(n + 2) .^{n} C_{0} . 2^{n + 1} - (n + 1)

^{n} C_{1}

2^{n} + n

^{n} C_{2}

2^{n - 1} +

= k (n + 1)

k

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