If the coefficients of x 7 and x 8 in the expansion of 2- x -3 n are equal- then the value of n isA- 15B- 55C- 56D- 45

The correct option is B 55(2+x3)^n=2^n(1+x6)^n ⇒ T_r+1=2^n·^nC_r·(x6)^r Now, coefficient of x^7= coefficient of x^8 ⇒ 2^n·^nC_7·16^7=2^n·^nC_8·16^8 ⇒ 6· ^nC_7= ...

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Standard XII

Mathematics

Progression

If the coeffi...

Question

If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of ${(2 + \frac{x}{3})}^{n}$ are equal, then the value of $n$ is

15

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45

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55

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56

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Solution

The correct option is C $55$
${(2 + \frac{x}{3})}^{n} = 2^{n} {(1 + \frac{x}{6})}^{n}$
$\Rightarrow T_{r + 1} = 2^{n} \cdot^{n} C_{r} \cdot {(\frac{x}{6})}^{r}$
Now, coefficient of $x^{7} =$ coefficient of $x^{8}$
$\Rightarrow 2^{n} \cdot^{n} C_{7} \cdot \frac{1}{6^{7}} = 2^{n} \cdot^{n} C_{8} \cdot \frac{1}{6^{8}}$
$\Rightarrow 6 \cdot^{n} C_{7} =^{n} C_{8}$
$\Rightarrow n - 7 = 48$
$\Rightarrow n = 55$

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If the coefficients of x7 and x8 in the expansion of (2+x3)n are equal, then the value of n is

The correct option is C 55(2+x3)n=2n(1+x6)n ⇒Tr+1=2n⋅nCr⋅(x6)r Now, coefficient of x7= coefficient of x8 ⇒2n⋅nC7⋅167=2n⋅nC8⋅168 ⇒6⋅ nC7= nC8 ⇒n−7=48 ⇒n=55

If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of ${(2 + \frac{x}{3})}^{n}$ are equal, then the value of $n$ is