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Binomial Theorem question

2003 · Shift 0 · Q104
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Binomial Theorem question

2003 · Shift 0 · Q104

JEE MainMathematicsBinomial TheoremMCQ+4 / −1
If xxx is positive, the first negative term in the expansion of (1+x)27/5{\left( {1 + x} \right)^{27/5}}(1+x)27/5 is
  1. A
    6th term
  2. B
    7th term
  3. C
    5th term
  4. D
    8th term.
View written solutionFree

Correct answer: D

  1. General term in the binomial expansion

For

(1+x)27/5,(1+x)^{27/5},(1+x)27/5,

the general term is

Tr+1=(27/5r)xr,T_{r+1} = \binom{27/5}{r} x^r,Tr+1​=(r27/5​)xr,

where

(27/5r)=(275)(275−1)(275−2)⋯(275−(r−1))r!.\binom{27/5}{r} = \frac{\left(\frac{27}{5}\right)\left(\frac{27}{5}-1\right)\left(\frac{27}{5}-2\right)\cdots\left(\frac{27}{5}-(r-1)\right)}{r!}.(r27/5​)=r!(527​)(527​−1)(527​−2)⋯(527​−(r−1))​.

Since x>0x>0x>0, the sign of each term depends only on the sign of the coefficient

(275)(225)(175)(125)(75)(25)(−35)⋯\left(\frac{27}{5}\right)\left(\frac{22}{5}\right)\left(\frac{17}{5}\right)\left(\frac{12}{5}\right)\left(\frac{7}{5}\right)\left(\frac{2}{5}\right)\left(-\frac{3}{5}\right)\cdots(527​)(522​)(517​)(512​)(57​)(52​)(−53​)⋯
  1. Find when the coefficient first becomes negative

Let us check the factors:

  • 275>0\frac{27}{5} > 0527​>0
  • 275−1=225>0\frac{27}{5}-1 = \frac{22}{5} > 0527​−1=522​>0
  • 275−2=175>0\frac{27}{5}-2 = \frac{17}{5} > 0527​−2=517​>0
  • 275−3=125>0\frac{27}{5}-3 = \frac{12}{5} > 0527​−3=512​>0
  • 275−4=75>0\frac{27}{5}-4 = \frac{7}{5} > 0527​−4=57​>0
  • 275−5=25>0\frac{27}{5}-5 = \frac{2}{5} > 0527​−5=52​>0
  • 275−6=−35<0\frac{27}{5}-6 = -\frac{3}{5} < 0527​−6=−53​<0

The first negative factor appears when r=7r=7r=7, because in Tr+1T_{r+1}Tr+1​ the last factor is

275−(r−1).\frac{27}{5}-(r-1).527​−(r−1).

So we need

275−(r−1)<0\frac{27}{5}-(r-1) < 0527​−(r−1)<0

which gives

r−1>275=5.4.r-1 > \frac{27}{5} = 5.4.r−1>527​=5.4.

Thus the smallest integer satisfying this is

r−1=6  ⟹  r=7.r-1=6 \implies r=7.r−1=6⟹r=7.

Hence the first negative term is

T7+1=T8.T_{7+1} = T_8.T7+1​=T8​.
  1. Conclusion

Therefore, the first negative term is the 8th term.

So the correct option is D.

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