If the co-efficients of x7 and x8 in the expansion of 2- x3n are equal- then the value of n is

The general term in nbsp;(2+ x3)^n is T_r+1 = ^nC_r · (2)^n r·( x3 )^r ⇒ T_r+1 = ^nC_r · (2)^n r·( 13 )^r· x^r Coefficient of x^7= ^nC_7 · (2)^n 7·( 13 )^7 A ...

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Byju's Answer

Standard XI

Mathematics

Binomial Theorem

If the co-eff...

Question

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

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Solution

The general term in ${(2 + \frac{x}{3})}^{n}$ is
$T_{r + 1} =^{n} C_{r} \cdot (2)^{n - r} \cdot {(\frac{x}{3})}^{r}$
$\Rightarrow T_{r + 1} =^{n} C_{r} \cdot (2)^{n - r} \cdot {(\frac{1}{3})}^{r} \cdot x^{r}$

Coefficient of $x^{7} =^{n} C_{7} \cdot (2)^{n - 7} \cdot {(\frac{1}{3})}^{7}$
And coefficient of $x^{8} =^{n} C_{8} \cdot (2)^{n - 8} \cdot {(\frac{1}{3})}^{8}$
Now, according to the question

$^{n} C_{7} \cdot 2^{n - 7} \cdot \frac{1}{3^{7}} =^{n} C_{8} \cdot 2^{n - 8} \cdot \frac{1}{3^{8}}$
$\Rightarrow \frac{^{n} C_{8}}{^{n} C_{7}} = 6$
$\Rightarrow \frac{n - 7}{8} = 6$
$∴ n = 55$

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

The general term in (2+x3)n is Tr+1= nCr⋅(2)n−r⋅(x3)r ⇒Tr+1= nCr⋅(2)n−r⋅(13)r⋅xr Coefficient of x7= nC7⋅(2)n−7⋅(13)7 And coefficient of x8= nC8⋅(2)n−8⋅(13)8 Now, according to the question nC7⋅2n−7⋅137= nC8⋅2n−8⋅138 ⇒ nC8 nC7=6 ⇒n−78=6 ∴n=55

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