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Question

For all real numbers a,b and positive integer n prove that:
(a+b)n=nC0an+nC1an1b+nC1an2b2+..........+nCnbn.

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Solution

Given: (a+b)n=nc0an+nc1an1b+nc2an2b2+.......+ncnbn........(1)
Let us prove above result by principles of mathematical induction as follows:
Step(i): For n=1,
LHS=(a+b)1
=(a+b)
R.H.S=1c0a1+1c1a11b
=a+a0b
=a+(1)b
=a+b
Thus, LHS=RHS.
Therefore, the result is true for n=1.
Step(ii): Assume the result is true for n=k.
i.e., (a+b)k=kc0ak+kc1ak1b+kc2ak2b2+.......+kckbk..........(2)
Step(iii): We shall prove that the result is true for n=k+1.
i.e., we need to prove (a+b)k+1=k+1c0ak+1+k+1c1akb+k+1c2ak1b2+.......+k+1ck+1bk+1 ........(3)
Consider LHS=(a+b)k+1
=(a+b)(a+b)k
=(a+b)×(kc0ak+kc1ak1b+kc2ak2b2+.......+kckbk) [From equation (2)]
=kc0ak+1+kc1akb+kc2ak1b2+.......+kckabk+kc0akb+kc1ak1b2+kc2ak2b3+.......+kckbk+1
=kc0ak+1+(kc1+kc0)(akb)+(kc2+kc1)(ak1b2)+.......+(kck+kck1)abk+kckbk+1
=k+1c0ak+1+k+1c1(akb)+k+1c2(ak1b2)+.......+k+1ckabk+k+1ck+1bk+1 [k+1c0=1,(ncr+ncr1)=n+1cr,k+1ck+1=1]
=R.H.S
Therefore, the result is true for n=k+1.
Hence, by principle of mathematical induction, the given results is true for all positive integers.


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