Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem B. 3874. (January 2006)

B. 3874. Define the sequence an (n is a natural number) as follows:


a_0=2 \quad\text{and} \quad a_n=a_{n-1}- \frac{n}{(n+1)!}, \quad\text{if} \quad n>0.

Express an in terms of n.

(National Mathematics Competition for Secondary Schools, 2005)

(3 pont)

Deadline expired on February 15, 2006.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás: Állítjuk, hogy a_n=1+{1\over (n+1)!}. Ez n=0 esetén így van, ha pedig valamely n természetes számra a_n=1+{1\over (n+1)!}, akkor

a_{n+1}=a_n-{n+1\over (n+2)!}=1+{1\over (n+1)!}-{n+1\over (n+2)!}=
1+{1\over (n+2)!},

állításunk helyessége tehát következik a teljes indukció elvéből.


Statistics:

195 students sent a solution.
3 points:156 students.
2 points:3 students.
1 point:1 student.
0 point:28 students.
Unfair, not evaluated:7 solutions.

Problems in Mathematics of KöMaL, January 2006