119443
If \(|x|\lt 1\), the coefficient of \(c^n\) in the expansion of \(\left(1+x+x^2+x^3+\ldots \ldots\right)^2\) is
1 \(\mathrm{n}\)
2 \(n+1\)
3 \(n-1\)
4 \(n+2\)
Explanation:
B The given expansion \(\left(1+\mathrm{x}+\mathrm{x}^2+\mathrm{x}^3+\ldots \ldots\right)^2\) \(\mathrm{S}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}\) So, \(\left[1+2 x+3 x^2+4 x^3+\ldots \ldots . .(x+1) x^k \ldots . .\right]\) The coefficient of \(x^n\) in the expositor \(={ }^{n+2-1} C_{2-1}={ }^{n+1} C_1\) \(=(n+1)\)
SRM JEEE-2011
Binomial Theorem and its Simple Application
119460
If the coefficient of the \(5^{\text {th }}\) term is the numerically the greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\), then the positive integral value of \(n\) is
1 10
2 9
3 8
4 7
Explanation:
C The middle term is the numerically greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\) i.e. \({ }^n C_{\mathrm{r}}\) is maximum at \(\mathrm{r}=\mathrm{n} / 2\) when \(\mathrm{n}\) is even \(=\frac{\mathrm{n}+1}{2}\) when \(\mathrm{n}\) is odd The given coefficient of \(5^{\text {th }}\) term is the greatest therefore \(5^{\text {th }}\) term is the middle term So, there are total 9 terms Hence, \(\mathrm{n}=8\)
JCECE-2016
Binomial Theorem and its Simple Application
119500
The coefficient of \(x^4\) in the expansion of \((1+x+\) \(x^2+x^3\) ) is
119443
If \(|x|\lt 1\), the coefficient of \(c^n\) in the expansion of \(\left(1+x+x^2+x^3+\ldots \ldots\right)^2\) is
1 \(\mathrm{n}\)
2 \(n+1\)
3 \(n-1\)
4 \(n+2\)
Explanation:
B The given expansion \(\left(1+\mathrm{x}+\mathrm{x}^2+\mathrm{x}^3+\ldots \ldots\right)^2\) \(\mathrm{S}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}\) So, \(\left[1+2 x+3 x^2+4 x^3+\ldots \ldots . .(x+1) x^k \ldots . .\right]\) The coefficient of \(x^n\) in the expositor \(={ }^{n+2-1} C_{2-1}={ }^{n+1} C_1\) \(=(n+1)\)
SRM JEEE-2011
Binomial Theorem and its Simple Application
119460
If the coefficient of the \(5^{\text {th }}\) term is the numerically the greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\), then the positive integral value of \(n\) is
1 10
2 9
3 8
4 7
Explanation:
C The middle term is the numerically greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\) i.e. \({ }^n C_{\mathrm{r}}\) is maximum at \(\mathrm{r}=\mathrm{n} / 2\) when \(\mathrm{n}\) is even \(=\frac{\mathrm{n}+1}{2}\) when \(\mathrm{n}\) is odd The given coefficient of \(5^{\text {th }}\) term is the greatest therefore \(5^{\text {th }}\) term is the middle term So, there are total 9 terms Hence, \(\mathrm{n}=8\)
JCECE-2016
Binomial Theorem and its Simple Application
119500
The coefficient of \(x^4\) in the expansion of \((1+x+\) \(x^2+x^3\) ) is
119443
If \(|x|\lt 1\), the coefficient of \(c^n\) in the expansion of \(\left(1+x+x^2+x^3+\ldots \ldots\right)^2\) is
1 \(\mathrm{n}\)
2 \(n+1\)
3 \(n-1\)
4 \(n+2\)
Explanation:
B The given expansion \(\left(1+\mathrm{x}+\mathrm{x}^2+\mathrm{x}^3+\ldots \ldots\right)^2\) \(\mathrm{S}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}\) So, \(\left[1+2 x+3 x^2+4 x^3+\ldots \ldots . .(x+1) x^k \ldots . .\right]\) The coefficient of \(x^n\) in the expositor \(={ }^{n+2-1} C_{2-1}={ }^{n+1} C_1\) \(=(n+1)\)
SRM JEEE-2011
Binomial Theorem and its Simple Application
119460
If the coefficient of the \(5^{\text {th }}\) term is the numerically the greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\), then the positive integral value of \(n\) is
1 10
2 9
3 8
4 7
Explanation:
C The middle term is the numerically greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\) i.e. \({ }^n C_{\mathrm{r}}\) is maximum at \(\mathrm{r}=\mathrm{n} / 2\) when \(\mathrm{n}\) is even \(=\frac{\mathrm{n}+1}{2}\) when \(\mathrm{n}\) is odd The given coefficient of \(5^{\text {th }}\) term is the greatest therefore \(5^{\text {th }}\) term is the middle term So, there are total 9 terms Hence, \(\mathrm{n}=8\)
JCECE-2016
Binomial Theorem and its Simple Application
119500
The coefficient of \(x^4\) in the expansion of \((1+x+\) \(x^2+x^3\) ) is
NEET Test Series from KOTA - 10 Papers In MS WORD
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Binomial Theorem and its Simple Application
119443
If \(|x|\lt 1\), the coefficient of \(c^n\) in the expansion of \(\left(1+x+x^2+x^3+\ldots \ldots\right)^2\) is
1 \(\mathrm{n}\)
2 \(n+1\)
3 \(n-1\)
4 \(n+2\)
Explanation:
B The given expansion \(\left(1+\mathrm{x}+\mathrm{x}^2+\mathrm{x}^3+\ldots \ldots\right)^2\) \(\mathrm{S}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}\) So, \(\left[1+2 x+3 x^2+4 x^3+\ldots \ldots . .(x+1) x^k \ldots . .\right]\) The coefficient of \(x^n\) in the expositor \(={ }^{n+2-1} C_{2-1}={ }^{n+1} C_1\) \(=(n+1)\)
SRM JEEE-2011
Binomial Theorem and its Simple Application
119460
If the coefficient of the \(5^{\text {th }}\) term is the numerically the greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\), then the positive integral value of \(n\) is
1 10
2 9
3 8
4 7
Explanation:
C The middle term is the numerically greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\) i.e. \({ }^n C_{\mathrm{r}}\) is maximum at \(\mathrm{r}=\mathrm{n} / 2\) when \(\mathrm{n}\) is even \(=\frac{\mathrm{n}+1}{2}\) when \(\mathrm{n}\) is odd The given coefficient of \(5^{\text {th }}\) term is the greatest therefore \(5^{\text {th }}\) term is the middle term So, there are total 9 terms Hence, \(\mathrm{n}=8\)
JCECE-2016
Binomial Theorem and its Simple Application
119500
The coefficient of \(x^4\) in the expansion of \((1+x+\) \(x^2+x^3\) ) is
119443
If \(|x|\lt 1\), the coefficient of \(c^n\) in the expansion of \(\left(1+x+x^2+x^3+\ldots \ldots\right)^2\) is
1 \(\mathrm{n}\)
2 \(n+1\)
3 \(n-1\)
4 \(n+2\)
Explanation:
B The given expansion \(\left(1+\mathrm{x}+\mathrm{x}^2+\mathrm{x}^3+\ldots \ldots\right)^2\) \(\mathrm{S}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}=\left(\frac{1}{1-\mathrm{x}}\right)^2=(1-\mathrm{x})^{-2}\) So, \(\left[1+2 x+3 x^2+4 x^3+\ldots \ldots . .(x+1) x^k \ldots . .\right]\) The coefficient of \(x^n\) in the expositor \(={ }^{n+2-1} C_{2-1}={ }^{n+1} C_1\) \(=(n+1)\)
SRM JEEE-2011
Binomial Theorem and its Simple Application
119460
If the coefficient of the \(5^{\text {th }}\) term is the numerically the greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\), then the positive integral value of \(n\) is
1 10
2 9
3 8
4 7
Explanation:
C The middle term is the numerically greatest coefficient in the expansion of \((1-x)^{\mathrm{n}}\) i.e. \({ }^n C_{\mathrm{r}}\) is maximum at \(\mathrm{r}=\mathrm{n} / 2\) when \(\mathrm{n}\) is even \(=\frac{\mathrm{n}+1}{2}\) when \(\mathrm{n}\) is odd The given coefficient of \(5^{\text {th }}\) term is the greatest therefore \(5^{\text {th }}\) term is the middle term So, there are total 9 terms Hence, \(\mathrm{n}=8\)
JCECE-2016
Binomial Theorem and its Simple Application
119500
The coefficient of \(x^4\) in the expansion of \((1+x+\) \(x^2+x^3\) ) is