To solve the problem of finding the number of paths from 1 to 10 where each step increases the number by 1 or 2, we can use dynamic programming.

Key Insight:

Let (dp[n]) denote the number of paths from 1 to (n).

• To reach (n), you can either:
  1. Take a step of +1 from (n-1), or
  2. Take a step of +2 from (n-2).

Thus, the recurrence relation is:
(dp[n] = dp[n-1] + dp[n-2])

Base cases:

• (dp[1] = 1) (only one way to stay at 1).
• (dp[2] = 1) (only one path: 1→2).

Calculations:

(dp[3] = dp[2] + dp[1] = 1+1=2)
(dp[4] = dp[3]+dp[2] =2+1=3)
(dp[5] =3+2=5)
(dp[6] =5+3=8)
(dp[7] =8+5=13)
(dp[8] =13+8=21)
(dp[9] =21+13=34)
(dp[10] =34+21=55)

Answer: 55.

(\boxed{55})

作者声明:本文包含人工智能生成内容。