Find the middle term of (2x-x^2/4)^9
Answers
Since the power is 9, there are 10 terms in the expansion, so the middle terms in the expansion are 5th and 6th terms.
We know that, in the expansion of (a + b)ⁿ, the (r + 1)th term is given by,
T_(r + 1) = nCr · a^(n - r) · b^r
Here a = 2x and b = - x² / 4. Also n = 9. Then, 5th term,
T_5 = 9C4 · (2x)^(9 - 4) · (- x² / 4)^4
T_5 = 126 · 2^5 · x^5 · x^8 / (4^4)
T_5 = 63 x^(13) / 4
And, 6th term,
T_6 = 9C5 · (2x)^(9 - 5) · (- x² / 4)^5
T_6 = - 126 · 2^4 · x^4 · x^(10) / (4^5)
T_6 = - 63 x^(14) / 32
Hence the middle terms are found.
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Find the middle term of (2x-x^2/4)^9
1). there are two middle terms ( 5th and 6th ) as total terms are 9 .
2). In expansion of (a + b)^n , the (r + 1)th term is T_(r + 1) = nCr · a^(n - r) · b^r
We know a = 2x and b = (-x^(2))/4
3). therefore 5th term is T_5 = 9C4·(2x)^(9 - 4)·(-x^(2) / 4)^4
T_5 = 63 x^(13) / 4
4). similarly 6th term is T_6 = 9C5.(2x)^(9-5).(-x^(2)/4)^4
T_6 = - 63 x^(14) / 32