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यदि O विषम पदों का योग तथा E सम पदों का योग है, तो सिद्ध कीजिये - (i) `O ^ 2 - E^2= ( x ^ 2 - a ^ 2 ) ^n ` (ii) ` 4OE = ( x + a ) ^(2n) - ( x -a ) ^ (2n ) ` (iii) `2(O^ 2 + E^ 2 ) = (x + a ) ^ (2n) + ( x - a)^ (2n) ` |
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Answer» ` (x + a ) ^n = ""^ n C _ 0 x ^n a ^0 + ""^n C _ 1 x ^( n - 1) a ^1 + ""^n C _ 2 x ^( n - 2 ) a ^ 2 + ... + ""^n C _ ( n - 1 ) x a ^( n - 1 ) + ""^n C _ n a ^n ` ` rArr (x + a )^n= (""^n C _ 0 x ^n a ^0 + ""^n C _ 2 x ^ (n - 2 ) a ^ 2 + ... ) + (""^n C _ 1 x ^(n - 1 ) a ^1 + ""^n C _ 3 x ^(n - 3 ) a ^ 3 + ... ) ` ` rArr ( x + a) ^n = O + E " " ` ...(i) तथा ` ( x - a ) ^n = ""^n C _ 0 x^ ( n ) - ""^n C _ 1 x ^ ( n - 1 ) a ^ 1 + ""^n C _2 x ^ (n - 2 ) a ^ 2 - "" ^n C _ 3 x ^(n - 3 ) a ^ 3 + ... + ""^n C _ ( n - 1 ) x ( - 1 ) ^(n - 1 ) a ^(n -1 ) a + ""^n C _ n ( - 1 ) ^n a ^n ` ` rArr (x - a ) ^n = (""^n C _ 0 x ^n + ""^n C _ 2 x ^(n - 2 ) a ^2 + ... ) - ( ""^n C _ 1 x ^(n - 1 ) a ^ 1 ""^n C _ 3 x ^(n -3 ) a ^ 3 +... ) ` ` rArr ( x -a ) ^n =O - E " " ` ... (iii) समीकरण (i ) व (ii ) की गुना करने पर, (i) ` (x + a)^n (x - a ) ^n = (O + E) ( O - E) rArr (x ^ 2 - a ^ 2 ) ^n = O^ 2 - E^2 ` (ii) ` 4 (OE) = [ ( O + E) ^ 2 - (O - E) ^ 2 ] ` ` 4 (OE) =[ ( x + a ) ^n] ^ 2 - [ (x - a )^n ] ^ 2 ` ` 4 (OE) = (x + a ) ^(2n ) ` (iii) समीकरण (i ) व (ii ) को वर्ग करके जोड़ने पर, ` (x + a ) ^(2n ) + (x - a ) ^(2n) = (O +E) ^ 2 + (O - E) ^ 2 = 2 (O^ 2 + E^2 )` |
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