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Prove that C_0 + 2. C_1 + 4. C_2 + 8. C_3 + ...... + 2^n . C_n

Question

Prove that $C_{0} + 2. C_{1} + 4. C_{2} + 8. C_{3} + ...... + 2^{n} . C_{n} = 3^{n}$ .

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Prove that

C_{0} + 2. C_{1} + 4. C_{2} + 8. C_{3} + ...... + 2^{n} . C_{n} = 3^{n}

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Text solutionVerified

Fact:

(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{3} x^{3} + ............ +^{n} C_{n} x^{n}

Substitute

x = 2

in above identity

(1 + 2)^{n} =^{n} C_{0} +^{n} C_{1} (2) +^{n} C_{2} (2)^{2} +^{n} C_{3} (2)^{3} + ............ +^{n} C_{n} (2)^{n}

∴^{n} C_{0} + 2 \cdot^{n} C_{1} + 4 \cdot^{n} C_{2} + 8 \cdot^{n} C_{3} + ........ + 2^{n} \cdot^{n} C_{n} = 3^{n}

Hence proved

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