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  • In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that (i) O^2– E^2= (x^2– a^2)n (ii) 4OE = (x + a)^2n- (x - a)^2n

In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E.

Then prove that

(i) O2– E2= (x2– a2)n                                          (ii) 4OE = (x + a)2n- (x - a)2n

Answers (1)

(i) (x+a)n= nC0xn + nC1xn-1 a1 + nC2xn-2 a2 + nC3xn-3 a3 + ….. + nCn an

Now, sum of odd terms,

O = nC0xn+ nC2xn-2 a2 + ……. 

& sum of even terms,

E = nC1xn-1 a1 + nC3xn-3 a3 + …..

Now, (x+a)n = O+E   &   (x-a)n = O – E

Thus, (O + E)(O – E) = (x + a)n(x – a)n

Thus, O2 – E2 = (x2 – a2)n

(ii)Here,

4OE = (O + E)2 – (O – E)2

               = [(x+a)n]2 – [(x-a)n]2

               = (x+a)2n – (x-a)2n
 

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