Let a-31-223-1 and for all n-3- let fn- nC0- an-1- nC1- an-2- nC2- an-3-1n-1- a0- If the value of f2007-f2008-3k where k - N- then the value of k is

F(n)= ^nC_0a^n 1 ^nC_1a^n 2+ ^nC_2a^n 3+…+ ( 1)^n 1 ^nC_n 1a^0 =1a( ^nC_0a^n ^nC_1a^n 1+ ^nC_2a^n 2+…+ ( 1)^n 1 ^nC_n 1a) =1a((a 1)^n ( 1)^n ^nC_n) =1a((3 ...

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Byju's Answer

Standard XII

Mathematics

Binomial Expression

Let a=31/223+...

Question

Let $a = 3^{1 / 223} + 1$ and for all $n \geq 3$ , let $f (n) =^{n} C_{0} \cdot a^{n - 1} -^{n} C_{1} \cdot a^{n - 2} +^{n} C_{2} \cdot a^{n - 3} - \dots + (- 1)^{n - 1} \cdot a^{0}$ . If the value of $f (2007) + f (2008) = 3^{k}$ where $k \in N$ , then the value of $k$ is

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Solution

$f (n) =^{n} C_{0} a^{n - 1} -^{n} C_{1} a^{n - 2} +^{n} C_{2} a^{n - 3} + \dots + (- 1)^{n - 1}^{n} C_{n - 1} a^{0}$

$= \frac{1}{a} (^{n} C_{0} a^{n} -^{n} C_{1} a^{n - 1} +^{n} C_{2} a^{n - 2} + \dots + (- 1)^{n - 1}^{n} C_{n - 1} a)$

$= \frac{1}{a} ((a - 1)^{n} - (- 1)^{n}^{n} C_{n})$

$= \frac{1}{a} ((3^{n / 223} - (- 1)^{n}^{n} C_{n}))$

$\Rightarrow f (x) = \frac{3^{n / 223} - (- 1)^{n}}{3^{1 / 223} + 1}$

$f (2007) = \frac{3^{2007 / 223} + 1}{3^{1 / 223} + 1}$

$f (2008) = \frac{3^{2008 / 223} - 1}{3^{1 / 223} + 1}$

$f (2007) + f (2008) = \frac{3^{2007 / 223} + 3^{2008 / 223}}{3^{1 / 223} + 1}$

$= \frac{3^{9} + 3^{9 + 1 / 223}}{3^{1 / 223} + 1}$

$= 3^{9} (\frac{3^{1 / 223} + 1}{3^{1 / 223} + 1}) = 3^{9}$

$\Rightarrow 3^{9} = 3^{k}$
Then, $k = 9$

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Let a=31/223+1 and for all n≥3, let f(n)= nC0⋅an−1− nC1⋅an−2+ nC2⋅an−3−…+(−1)n−1⋅a0. If the value of f(2007)+f(2008)=3k where k∈N, then the value of k is

Let $a = 3^{1 / 223} + 1$ and for all $n \geq 3$ , let $f (n) =^{n} C_{0} \cdot a^{n - 1} -^{n} C_{1} \cdot a^{n - 2} +^{n} C_{2} \cdot a^{n - 3} - \dots + (- 1)^{n - 1} \cdot a^{0}$ . If the value of $f (2007) + f (2008) = 3^{k}$ where $k \in N$ , then the value of $k$ is