Let for the 9th term in the binomial expansion of (3 + 6x)n, in the increasing powers of 6x, to be the greatest for x=32, the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k + n0 is equal to:

(3+6x)n=nC03n+nC13n−1(6x)1+…Tr+1nCr3n−r⋅(6x)r=nCr3n−r⋅6r⋅xr=nCr3n−r⋅3r⋅2r⋅32r=nCr3n⋅3r for x=32T9 is greatest of x=32So, T9 > T10 and T9 > T8(concept of numerically greatest term)Here, T9T10>1 and T9T8>1⇒ nC83n⋅38 nC93n⋅39>1 and nC83n⋅38 nC73n⋅37>1and nC8 nC7>13and n−78>13⇒293<n<11⇒n=10=n0So, in (3+6x)n for n=n0=10i.e., in (3+6x)10, here Tr+1=10Cr310−r6rxrT7=10C634⋅66⋅x6=210⋅310⋅26x6T4=10C33763x3=120⋅310⋅23x3Ratio of coefficient of x6 and coefficient of x3 = k∴k=210⋅31026120.310⋅23=74×23=14So, k+n0=14+10=24

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Let for the 9^th term in the binomial expansion of (3 + 6x)ⁿ, in the increasing powers of 6x, to be the greatest for $x = \frac{3}{2}$ , the least value of n is n₀. If k is the ratio of the coefficient of x⁶ to the coefficient of x³, then k + n₀ is equal to:

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Detailed Solution

$\begin{array}{l} (3 + 6 x)^{n} =^{n} C_{0} 3^{n} +^{n} C_{1} 3^{n - 1} (6 x)^{1} + \dots \\ {T_{r + 1}}^{n} C_{r} 3^{n - r} \cdot (6 x) r =^{n} C_{r} 3^{n - r} \cdot 6^{r} \cdot x^{r} \\ =^{n} C_{r} 3^{n - r} \cdot 3^{r} \cdot 2^{r} \cdot {(\frac{3}{2})}^{r} =^{n} C_{r} 3^{n} \cdot 3^{r} [for x = \frac{3}{2}] \end{array}$

T₉ is greatest of $x = \frac{3}{2}$

So, T₉ > T₁₀ and T₉ > T₈

(concept of numerically greatest term)

Here, $\frac{T_{9}}{T_{10}} > 1 and \frac{T_{9}}{T_{8}} > 1$

$\Rightarrow \frac{^{n} C_{8} 3^{n} \cdot 3^{8}}{^{n} C_{9} 3^{n} \cdot 3^{9}} > 1 and \frac{^{n} C_{8} 3^{n} \cdot 3^{8}}{^{n} C_{7} 3^{n} \cdot 3^{7}} > 1$

and $\frac{^{n} C_{8}}{^{n} C_{7}} > \frac{1}{3}$

and $\frac{n - 7}{8} > \frac{1}{3}$

$\Rightarrow \frac{29}{3} < n < 11 \Rightarrow n = 10 = n_{0}$

So, $in (3 + 6 x)^{n} for n = n_{0} = 10$

i.e., $in (3 + 6 x)^{10}, here T_{r + 1} =^{10} C_{r} 3^{10 - r} 6^{r} x^{r}$

$\begin{array}{l} T_{7} =^{10} C_{6} 3^{4} \cdot 6^{6} \cdot x^{6} = 210 \cdot 3^{10} \cdot 2^{6} x^{6} \\ T_{4} =^{10} C_{3} 3^{7} 6^{3} x^{3} = 120 \cdot 3^{10} \cdot 2^{3} x^{3} \end{array}$