Hurry tell this ques by writing all statements solve all - Maths - Mathematical Induction and Binomial Theorem

Question

Hurry tell this ques by writing all statements solve all

Manbar Singh · Accepted Answer

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We have, 1 +xn = C0n + C1nx + C2nx2 + 
+Cnn xn     
1Also, x + 1n = C0n xn+ C1nxn-1 + C2nxn-2 + 
+Cnn      
2Multiply 1 and 2, we get1+x2n = C0n + C1nx + C2nx2 + 
+Cnn xnC0n xn+ C1nxn-1 + C2nxn-2 + 
+Cnn    
3Now, we know that in the expansion of 1+x2n,Tr+1 = Cr2n 12n-r xr = Cr2n xrCoefficient of xn = Cn2nNow, in 3, equating the coefficients of xn both sides, we getCn2n = C0n2 + C1n2 +  C2n2 + 
 +  Cnn2⇒C02 + C12 +  C22 + 
 +  Cn2 = Cn2n 
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