If 3rd, 4th, 5th and 6th terms in the expansion of $(x + α)^{n}$ be respectively a,b,c and d. prove that $\frac{b^{2} - a c}{c^{2} - b d} = \frac{5 a}{3 c} .$

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Let the coefficient of 6th term of an expansion be a and b be the power.Expansion:(4x5−52x)9Find a×b.

The correct option is B 5040(r+1)th term is tr+1=nCran−rbrSo, for 6th term, r=5.∴t6=t5+1=9C5(−1)5(4x5)9−5(52x)5∴t6=−5040xSo, a=−5040,b=−1Ans-Option B.

Let the coefficient of $6^{t h}$ term of an expansion be $a$ and $b$ be the power.
Expansion: ${(\frac{4 x}{5} - \frac{5}{2 x})}^{9}$
Find $a \times b$ .

The correct option is B $5040$
$(r + 1)^{t h}$ term is $t_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$
So, for $6^{t h}$ term, $r = 5$ .
$∴ t_{6} = t_{5 + 1} =^{9} C_{5} (- 1)^{5} (\frac{4 x}{5})^{9 - 5} (\frac{5}{2 x})^{5}$
$∴ t_{6} = \frac{- 5040}{x}$
So, $a = - 5040, b = - 1$
Ans-Option $B$ .