If n - N and if 1 - 4x - 4x2n - -r - 02narxr - where ao-a1-a2-a2n are real numbers- The value of a2n - 1 is -

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

Byju's Answer

Standard XII

Mathematics

Middle Terms

If n ∈ N an...

Question

If $n \in N$ and if ${(1 + 4 x + 4 x^{2})}^{n} = 2 n \sum r = 0 a_{r} x^{r}$ , where $a_{o}, a_{1}, a_{2}, . . . . . . ., a_{2 n}$ are real numbers. The value of $a_{2 n - 1}$ is -

2^{2 n}

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

n . 2^{2 n}

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

(n - 1) 2^{2 n}

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

(n + 1) 2^{2 n}

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

Open in App

Solution

The correct option is B $n . 2^{2 n}$
Solution -

$= (1 + 4 x + 4 x^{2})^{n}$
$= (2 x + 1)^{2 n} = (1 + 2 x)^{2 n}$

$=^{2 n} C_{0} +^{2 n} C_{1} (2 x) - - - - - - - +^{2 n} C_{2 n} (2 x)^{2 n}$
$a_{n - 1} =^{2 n} C_{2 n - 1} (2 x)^{2 n - 1}$

$a_{n - 1} = 2 n \times 2^{2 n - 1}$

$a_{n - 1} = n {.2}^{2 n}$

B is correct

Suggest Corrections

Similar questions

Q. If

n \in N

and if

(1 + 4 x + 4 x^{2})^{n} = \sum_{r = 0}^{2 n} a_{r} x^{r}

, where

a_{0}, a_{1}, a_{1}, a_{2}, . . . . . . . . ., a_{2 n}

are real numbers.
The value of

2 \sum_{r = 0}^{n} a_{2 r}

, is

Q. If

n \in N

and if

{(1 + 4 x + 4 x^{2})}^{n} = 2 n \sum r = 0 a_{r} X^{r},

, where

a_{o}, a_{1}, a_{2}, . . . . . . ., a_{2 n}

are real numbers. The value of

n \sum r = 0 a_{2 r}

, is

Q. If

n

is the positive integer and if

(1 + 4 x + 4 x^{2})^{n} = 2 n \sum r = 0 a_{r} x^{r}

, where

a_{r}

are real numbers. The value of

a_{2 n - 1}

Q. If

a_{r} > 0; \forall r, n \in N

and

a_{1}, a_{2}, a_{3}, . . . . . a_{2 n}

are in A.P, then

\frac{a_{1} + a_{2 n}}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{a_{2} + a_{2 n - 1}}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \frac{a_{3} + a_{2 n - 2}}{\sqrt{a_{3}} + \sqrt{a_{4}}} + . . . + \frac{a_{n} + a_{n + 1}}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}} =

Given that a_i $>$ 0 and i belongs to a set of natural numbers. If $a_{1}, a_{2}, a_{3} . . . . . a_{2 n}$ are in AP, then find the value of $\frac{a_{1} + a_{2 n}}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{a_{2} + a_{2 n} - 1}{{\sqrt{a}}_{2} + \sqrt{a_{3}}} . . . . . . . . . . . . + \frac{a_{n} + a_{n + 1}}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}}$

Join BYJU'S Learning Program

If n∈N and if (1+4x+4x2)n=2n∑r=0arxr , where ao,a1,a2,.......,a2n are real numbers. The value of a2n−1 is -

The correct option is B n.22nSolution -=(1+4x+4x2)n=(2x+1)2n=(1+2x)2n=2nC0+2nC1(2x)−−−−−−−+2nC2n(2x)2nan−1=2nC2n−1(2x)2n−1an−1=2n×22n−1an−1=n.22nB is correct

If $n \in N$ and if ${(1 + 4 x + 4 x^{2})}^{n} = 2 n \sum r = 0 a_{r} x^{r}$ , where $a_{o}, a_{1}, a_{2}, . . . . . . ., a_{2 n}$ are real numbers. The value of $a_{2 n - 1}$ is -