If (1+x)2n=a0+a1x....+a2nx2n, then

Question

If (1+x)2n=a0​+a1​x....+a2n​x2n, then

Filo · Accepted Answer

(1+x) 2n+(1−x) 2n=2(a0​+a2​x2+a4​x4+...+a2n​x2n)For x=1,      222n​=(a0​+a2​+a4​+...+a2n​)(1+x) 2n=(a0​+a1​x+a2​x2+...+a2n​x2n)                                           ...(i)For x=1,    22n=a0​+a1​+a2​+....+a2n​Hence, (a0​+a2​+a4​+...+a2n​)= 1/2(a0​+a1​+a2​+....+a2n​)So, A is correct.Now, x2n(1+x)2n​=x2n(a0​+a1​x+...+a2n​x2n)​So, (1+x1​ ) 2n=x2nao​​+x2n−1a1​​+....+x0a2n​​                                              ...(ii)Putting x=x1​  in (i), (1+x1​ ) 2n=a0​+xa1​​+x2a2​​+...+x2na2n​​                                                   ...(iii)On comparing coefficients from (ii) and (iii),   a2n​=a0​,   a2n−1​=a1​.... and so on...So, we can say,   a2n−r​=ar​.  for 2n≥r≥0For, r=n+3,   an−3​=an+3​                       So, C is correct.But for r=n, we find  B is incorrect.

Question Text	If $(1 + x)^{2 n} = a_{0} + a_{1} x .... + a_{2 n} x^{2 n}$ , then
Answer Type	Text solution:1
Upvotes	150

If $(1 + x)^{2 n} = a_{0} + a_{1} x .... + a_{2 n} x^{2 n}$ , then

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If (1+x)2n=a0​+a1​x....+a2n​x2n, then

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If $(1 + x)^{2 n} = a_{0} + a_{1} x .... + a_{2 n} x^{2 n}$ , then