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  • Find the middle term (terms) in the expansion of (i). (x/a - a/x)^10 (ii) (3x-x^3/6)^9

Find the middle term (terms) in the expansion of

i) \left ( \frac{x}{a}-\frac{a}{x} \right )^{10}                     ii) \left ( 3x-\frac{x^{3}}{6} \right )^{9}

Answers (1)

(i) Given: (x/a – a/x)10

Where, n = 10(even).

The middle term = (10/2 + 1)th term, viz., the sixth term

Thus, T6 = T5+1 10C5 (x/a)10-5 (-a/x)5

                 = - 10C5 (x/a)5(a/x)5

                 = - 10 x 9 x 8 x 7 x 6 x 5!/ 5! x 4 x 3 x 2 x 1 (x/a)5(x/a)-5

                 = -252

(ii) Given: (3x – x3/6)9

Where, n = 9 (odd)

Now, here are two middle terms-

(9+1/2)th ,viz., the tenth term

& (9+1/2  + 1)th, viz., the sixth term

Thus, T5 = T4+1

                 = 9C4 (3x)9-4 (-x3/6)4

                 = 9x8x7x6x5!/4x3x2x1x5!  . 35x5x126-4

                 = 189/8 x17

Now,

T6 = T5+1

    = 9C5(3x)9-5 (-x3/6)5

    = - 9 x 8 x 7 x 6 x 5!/ 5! x 4 x 3 x 2 x 1 . 34.x4.x15.6-4

                 = - 21/16 x19

 

Posted by

infoexpert22

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