Question:
(1+ x)^n=a0 +a1x+ a2x^2 ..+.......... +anx^n then find i)ao-a2 +a4-a6 .......... ii)a1-a3+ a5-a7 ......... iii)a0+ a3+ a6 +a9+-------
Answer:
Given, (1 + x)n = a0 + a1 x + a2 x2 +.......... + an xn .............1
Put x = 1, we get
(1 + 1)n = a0 + a1 + a2 +.......... + an
=> 2n = a0 + a1 + a2 +.......... + an ...........2
Again, put x = -1, we get
(1 - 1)n = a0 - a1 + a2 -.......... + (-1)n an
=> 0 = a0 - a1 + a2 -.......... + (-1)n an ..............3
1. Now, add equation 2 and 3, we get
2n = (a0 + a1 + a2 +.......... + an ) + (a0 - a1 + a2 -.......... + (-1)n an )
=> 2n = a0 + a1 + a2 +.......... + an + a0 - a1 + a2 -.......... + (-1)n an
=> 2n = 2(a0 + a2 +.......... + an )
=> 2n /2 = a0 + a2 +.......... + an
=> a0 + a2 +.......... + an = 2n-1
2. Again subtract equation 2 and 3 we get
2n = (a0 + a1 + a2 +.......... + an ) - (a0 - a1 + a2 -.......... + (-1)n an )
=> 2n = a0 + a1 + a2 +.......... + an - a0 + a1 - a2 -.......... + (-1)n an
=> 2n = 2(a1 + a3 +.......... + an-1 )
=> 2n /2 = a1 + a3 +.......... + an-2
=> a1 + a3 +.......... + an-1 = 2n-1
3. The solution of 3rd part will be written soon.
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