Question

鉴于，

1*2^1 + 2*2^2 + 3*2^3 + 4*2^4 + ... d * 2^d

= sum(r * 2^r, r from 1 to d)

我们如何推断出以下解决方案？

= 2 (d-1) * 2^d + 2

谢谢

Answer 1

d 上的induction：

基本案例

d = 1
sum(r * 2^r, r from 1 to 1) = 1 * 2^1 = 1 * 2 = 2
2 * (1 - 1) * 2^1 + 2 = 2 * 0 * 2 + 2 = 0 + 2 = 2

归纳案例

我们假设归纳假设对于 d 是正确的：

sum(r * 2^r, r from 1 to d + 1) =
sum(r * 2^r, r from 1 to d) + [(d + 1) * 2^(d + 1)] =
2 * (d-1) * 2^d + 2 + [(d + 1) * 2^(d + 1)] =
(d - 1) * 2^(d + 1) + 2 + d * 2^(d + 1) + 2^(d + 1) =
d * 2^(d + 1) - 2^(d + 1) + 2 + d * 2^(d + 1) + 2^(d + 1) =
d * 2^(d + 1) + 2 + d * 2^(d + 1) =
2 * d * 2^(d + 1) + 2 (result 1)

和现在评估d + 1

的公式

2 (d-1) * 2^d + 2 = (substituting d + 1 for d)
2 * (d + 1 - 1) * 2^(d + 1) + 2 =
2 * d * 2^(d + 1) + 2 (result 2)

从而

2 * d * 2^(d + 1) + 2 (result 1) = 2 * d * 2^(d + 1) + 2 (result 2)

QED

Answer 2

我认为你可以通过归纳证明来证明这一点：

http://en.wikipedia.org/wiki/Proof_by_induction

如何计算以下公式

2 个答案: