Question

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Answer 1

Answer:

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Answer 2

QUESTION -

[ in Given, attached file]

ANSWER -

(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

@HarshPratapSingh