Câu hỏi 48
Chứng minh rằng
$$
2\sum_{k=0}^{n}\binom{2n}{k}=\binom{2n}{n}+2^{2n},\qquad n=1,2,\ldots
$$
Đồng thời chứng minh rằng
$$
2\sum_{k=0}^{n}\binom{2n}{k}(-1)^k=\binom{2n}{n}(-1)^n,\qquad n=1,2,\ldots
$$
$$
2\sum_{k=0}^{n}\binom{2n}{k}=\binom{2n}{n}+2^{2n},\qquad n=1,2,\ldots
$$
Đồng thời chứng minh rằng
$$
2\sum_{k=0}^{n}\binom{2n}{k}(-1)^k=\binom{2n}{n}(-1)^n,\qquad n=1,2,\ldots
$$