In the expansion of ( x^ 3/4 +1 x^ 1/2 -x^ 1/4 +1 - x-1 x+x^ 2/3 +x^ 1/3 )²⁰ , x 1 , in decreasing powers of x , the ratio of the 5^ th term from the beginning to the 5^ th term from the end is x^k . The value of k is:

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In the expansion of ( x^ 3/4 +1 x^ 1/2 -x^ 1/4 +1 - x-1 x+x^ 2/3 +x^ 1/3 )²⁰ , x > 1 , in decreasing powers of x , the ratio of the 5^ th term from the beginning to the 5^ th term from the end is x^k . The value of k is:

Quantrex Academy · Accepted Answer

Correct answer: B. 7

aligned & x^ 3/4 +1 x^ 1/2 -x^ 1/4 +1 - x-1 x+x^ 2/3 +x^ 1/3 & = (x^ 1/4 +1)(x^ 1/2 -x^ 1/4 +1) x^ 1/2 -x^ 1/4 +1 - (x^ 1/3 -1)(x^ 2/3 +x^ 1/3 +1) x^ 1/3 (x^ 2/3 +x^ 1/3 +1) & = (x^ 1/4 + 1) - x^ 1/3 - 1 x^ 1/3 & = (x^ 1/4 + 1) - (1 - x^ -1/3 ) & = x^ 1/4 + x^ -1/3 aligned The given expression becomes (x^ 1/4 + x^ -1/3 )²⁰ . The 5^ th term from the beginning is: T₅ = T_ 4+1 = ²⁰C₄ (x^ 1/4 )¹⁶ (x^ -1/3 )^4 = ²⁰C₄ x^4 x^ -4/3 = ²⁰C₄ x^ 8/3 The 5^ th term from the end in the expansion of (a+b)^n is the (n - 5 + 2)^ th

In the expansion of ( x^ 3/4 +1 x^ 1/2 -x^ 1/4 +1 - x-1 x+x^ 2/3 +x^ 1/3 )²⁰ , x > 1 , in decreasing powers of x , the ratio of the 5^ th term from the beginning to the 5^ th term from the end is x^k . The value of k is:

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