Question

Find the coefficient of:

(i)  x¹⁰ in the expansion of (2x² – 1/x)²⁰

(ii) x⁷ in the expansion of (x – 1/x²)⁴⁰

(iii) x^-15 in the expansion of (3x² – a/3x³)¹⁰

(iv) x⁹ in the expansion of (x² – 1/3x)⁹

Answer 1

(i)  x¹⁰ in the expansion of (2x² – 1/x)²⁰

Given as

(2x² – 1/x)²⁰

If  x¹⁰ occurs in the (r + 1)^th term in the given expression.

Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

(ii) x⁷ in the expansion of (x – 1/x²)⁴⁰

Given as

(x – 1/x²)⁴⁰

If x⁷ occurs at the (r + 1)^th term in the given expression.
Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

(iii) x^-15 in the expansion of (3x² – a/3x³)¹⁰

Given as

(3x² – a/3x³)¹⁰

If x⁻¹⁵ occurs at the (r + 1)^th term in the given expression.
Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

(iv) x⁹ in the expansion of (x² – 1/3x)⁹

Given as

(x² – 1/3x)⁹

If x⁹ occurs at the (r + 1)^th term in the above expression.

Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

For this term to contain x⁹, we must have:

18 − 3r = 9

3r = 18 – 9

3r = 9

r = 9/3

= 3