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

Question

Using Binomial Theorem, evaluate \((102)^5\).

Answer 1

Given:

∑_r=0ⁿ 3^r · ⁿC_r

Step 1: Use Binomial Theorem

According to binomial theorem:

(a + b)ⁿ = ∑_r=0ⁿ ⁿC_r a^n−r b^r

Step 2: Choose suitable values

Let a = 1 and b = 3

Then,

(1 + 3)ⁿ = ∑_r=0ⁿ ⁿC_r (1)^n−r (3)^r

Step 3: Simplify

(1 + 3)ⁿ = ∑_r=0ⁿ ⁿC_r 3^r

4ⁿ = ∑_r=0ⁿ 3^r · ⁿC_r

Final Result:

∑_r=0ⁿ 3^r · ⁿC_r = 4ⁿ

Solution and Explanation