Find the term in x2 and the term independent of x in the expansion of

1.	Find the term in x2 and the term independent of x in the expansion of (2x + 1/x)12.
Answer» \(\because\) General term in expansion of (x + a)ⁿ is T_r+1 = ⁿC_r x^n-ra^r \(\therefore\) General term in expansion of (2x + 1/x)¹² is T_r+1 = ¹²C_r (2x)^n-r(1/x)^r = ¹²C_r 2^n-r x^n-r-r = ¹²C_r 2^n-r x^n-2r = ¹²C_r 2^12-r x^12-2r---(1) (\(\because\) n = 12) For term in x², we have 12 - 2r = 2 ⇒ 2r = 12 - 2 = 10 ⇒ r = 10/2 = 5 \(\therefore\) Term in x² in the expansion of (2x + 1/x)¹² is T₆ = ¹²C₅2^12-5 x² (from (1)) = ¹²C₅2⁷ x² For constant term (term independent of x), we have 12 - 2r = 0 ⇒ 2r = 12 ⇒ r = 12/2 = 6 \(\therefore\) constant term (term independent of x) in the expansion of (2x + 1/x)¹² T₇ = ¹²C₆2^12-6x⁰ (From (1)) = ¹²C₆.2⁶

Find the term in x2 and the term independent of x in the expansion of (2x + 1/x)12.

Answer»

\(\because\) General term in expansion of (x + a)ⁿ is

T_r+1 = ⁿC_r x^n-ra^r

\(\therefore\) General term in expansion of (2x + 1/x)¹² is

T_r+1 = ¹²C_r (2x)^n-r(1/x)^r

= ¹²C_r 2^n-r x^n-r-r

= ¹²C_r 2^n-r x^n-2r

= ¹²C_r 2^12-r x^12-2r---(1) (\(\because\) n = 12)

For term in x², we have

12 - 2r = 2

⇒ 2r = 12 - 2 = 10

⇒ r = 10/2 = 5

\(\therefore\) Term in x² in the expansion of (2x + 1/x)¹² is

T₆ = ¹²C₅2^12-5 x² (from (1))

= ¹²C₅2⁷ x²

For constant term (term independent of x), we have

12 - 2r = 0

⇒ 2r = 12

⇒ r = 12/2 = 6

\(\therefore\) constant term (term independent of x) in the expansion of

(2x + 1/x)¹² T₇ = ¹²C₆2^12-6x⁰ (From (1))

= ¹²C₆.2⁶

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