**Engel 9.25.** The sequence is defined by , . Find the integer part of the sum

Note: This problem was taken from Arthur Engel’s *Problem-Solving Strategies.*

**Let’s work out the small values of :**

and are smaller than 1. is greater than 1. Looking at the recurrence relation, note that a term is bigger than the square of the previous terms. This means that **once our numbers go past 1, they increase very very quickly**. For example, compare this with the sequence . The sequence is going to grow even faster than this.

Now, noting that

this suggests that is going to converge very quickly as well, and to a small value.

**How can we use the recurrence relation to get the terms ?** Well, is already lurking in the RHS:

This is a big result! **We can turn the sum we are supposed to evaluate into a telescoping sum**, canceling many terms in the middle:

Hence, the integer part of the sum is less than 2. From our earlier analysis, we know that for large , is large, which means that is small, almost definitely less than 1, which means that the sum above would be equal to “1 point something”. We just need to show that

, or .

But that is (more or less obvious), we always have , which means that .

Thus, the integer part of the sum is **1**. **Done!**

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