If a-31-223-1 and fn-r-1n-1r-1 n Cr-1- an-r for all n - 3- then the value of f2007-f2008 is

F(n)=∑_r=1^n( 1)^r 1 ^n C_r 1· a^n r nbsp;= 1 a∑_r=1^n( 1)^r 1 ^n C_r 1· a^n r+1 = 1 a[ ^n C_0 a^n ^n C_1 a^n 1+⋯+( 1)^n 1 ^nC_n 1 a^1] =1a[(a 1)^n ( 1)^n ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XIII

Mathematics

Sum of Coefficients of All Terms

If a=31/223+1...

Question

If $a = 3^{1 / 223} + 1$ and $f (n) = n \sum r = 1 (- 1)^{r - 1}^{n} C_{r - 1} \cdot a^{n - r}$ for all $n \geq 3$ , then the value of $f (2007) + f (2008)$ is

3^{6}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3^{9}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

3^{12}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3^{15}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $3^{9}$
$f (n) = n \sum r = 1 (- 1)^{r - 1}^{n} C_{r - 1} \cdot a^{n - r} = \frac{1}{a} n \sum r = 1 (- 1)^{r - 1}^{n} C_{r - 1} \cdot a^{n - r + 1} = \frac{1}{a} [^{n} C_{0} a^{n} -^{n} C_{1} a^{n - 1} + \dots + (- 1)^{n - 1}^{n} C_{n - 1} a^{1}] = \frac{1}{a} [(a - 1)^{n} - (- 1)^{n} C_{n}] = \frac{1}{a} [3^{n / 223} - (- 1)^{n}] \Rightarrow f (n) = \frac{3^{n / 223} - (- 1)^{n}}{(3^{1 / 223} + 1)}$
Now,
$f (2007) + f (2008) = \frac{3^{2007 / 223} + 3^{2008 / 223}}{3^{1 / 223} + 1} = \frac{3^{9} + 3^{2008 / 223}}{3^{1 / 223} + 1} = 3^{9} [\frac{1 + 3^{1 / 223}}{3^{1 / 223} + 1}] = 3^{9}$

Suggest Corrections

Similar questions

Q. If

a = 3^{1 / 223} + 1

and

f (n) = n \sum r = 1 (- 1)^{r - 1}^{n} C_{r - 1} \cdot a^{n - r}

for all

n \geq 3

, then the value of

f (2007) + f (2008)

Q. Let

a = 3^{\frac{1}{223}} + 1

and let

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

. If the values of

f (2007) + f (2008) = 9 k

where

k ϵ N

, then find

k

Q. Let

a_{0} = 1

and

a_{n + 1} = a_{0} \cdot \cdot \cdot a_{n} + 4,

for all

n \geq 0.

Then

a_{n} - \sqrt{a_{n + 1}} =

for all

n \geq 1.

Q. If

f (n) = {cot}^{- 1} (n + 3) - 2 {cot}^{- 1} (n + 1) + {cot}^{- 1} (n - 1)

for

n \in N,

then

\infty \sum n = 1 f (n)

is equal to

Q. The value of

lim x \to \infty \frac{1 \cdot n \sum r = 1 r + 2 \cdot n - 1 \sum r = 1 r + 3 \cdot n - 2 \sum r = 1 r + . . . . + n \cdot 1}{n^{4}}

Join BYJU'S Learning Program

Explore more

Sum of Coefficients of All Terms

Standard XIII Mathematics

Join BYJU'S Learning Program

If a=31/223+1 and f(n)=n∑r=1(−1)r−1 nCr−1⋅an−r for all n≥3, then the value of f(2007)+f(2008) is

If $a = 3^{1 / 223} + 1$ and $f (n) = n \sum r = 1 (- 1)^{r - 1}^{n} C_{r - 1} \cdot a^{n - r}$ for all $n \geq 3$ , then the value of $f (2007) + f (2008)$ is