Question

Prove that \(\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n\) 

Answer 1

Given:

∑_r=0ⁿ 3^r · ⁿC_r

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Step 1: Use Binomial Theorem

According to binomial theorem:

(a + b)ⁿ = ∑_r=0^{n n}C_r a^n−r b^r

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Step 2: Choose suitable values

Let a = 1 and b = 3

Then,

(1 + 3)ⁿ = ∑_r=0^{n n}C_r (1)^n−r (3)^r

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Step 3: Simplify

(1 + 3)ⁿ = ∑_r=0^{n n}C_r 3^r

4ⁿ = ∑_r=0ⁿ 3^r · ⁿC_r

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Final Result:

∑_r=0ⁿ 3^r · ⁿC_r = 4ⁿ