Try to solve this problem number 35 from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation.
Consider integers \({1,2, \ldots, 10}\). A particle is initially -at 1 . It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time?
Functional Equation
Equation
Answer : 81
Singapore Mathematical Olympiad
Challenges an Thrills - Pre - College Mathematics
If you got stuck into this problem we can start taking an expected number of steps to be \(g_{n}\). We need to remember at first the particle was in 1 then it will shift to the next step so for n no of position we can expressed it as n and n -1 where n = 2,3,4,........,100.
Now try the rest..............
Now let's continue after the last hint ............
Then \(g_{n+1} = \frac {1}{2} (1+g_{n} + g_{n+1} )+ \frac {1}{2}\)
which implies , \(g_{n+1} = g_{n} + 2\)
Now we know that,\(g_{2} = 1\). Then \(g_{3} = 3\), \(g_{4}= 5\),..................,\(g_{10}=17\)
\(g = g_{2}+g_{3}+g_{4}+....................+g_{10} = 1+3+.....................+17 = 81\)[ Answer]
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