Question

find the term independent of x in the expansion of (x-1/x^2) ^15​

Answer 1

Step-by-step explanation:

The given expression is

$(x - \frac{1}{ {x}^{2} } ) ^{15}$

Let r th term be the term independent of x

$t _{r + 1} = ( - 1)^{r} \: ^{15}c_{r}. {x}^{15 - r} . (\frac{1}{ {x}^{2} })^{r}$

$\implies \: t _{r + 1} = ( - 1) ^{r} \: ^{15}c_{r}. {x}^{15 - r} . {x}^{ - 2r}$

$\implies \: t _{r + 1} = ( - 1) ^{r} \: ^{15} \: c_{r} . {x}^{15 - r - 2r}$

$\implies \: t _{r + 1} = ( - 1) ^{r} \: ^{15}c_{r}. {x}^{15 - 3r}$

Now, to be independent of x, the power of x should be equal to 0

so,

$15 - 3r = 0$

$\implies \: r = 5$

So ,the term independent of x is

$\implies \: t _{6} = ( - 1) ^{5} \: ^{15}c_{5}$

$\implies \: t _{6} = ( - 1 ) \times \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1}$

$\implies \: t_{6} = - 3003$