A snail crawls from one tree to another, covering the same extra distance each day compared with the previous day. On the first and last days together it crawled 10 metres in total. The trees are 150 metres apart. How many days did the journey take?

Step-by-step solution

1. 1

   Identify the sequence. "The same extra distance each day" means the daily distances $a_1,a_2,\ldots,a_n$ form an arithmetic progression. The total distance is the sum of the progression, and the total is given as $150$ m.

2. 2

   Choose the form of the sum that matches the given data. The two standard formulas are

   $S_n=\frac{a_1+a_n}{2}\cdot n\qquad\text{and}\qquad S_n=\frac{2a_1+(n-1)d}{2}\cdot n$

   The problem supplies $a_1+a_n=10$ directly, so the first form needs no knowledge of $a_1$ or the common difference $d$ at all — that is what makes the problem solvable from so little data.

3. 3

   Substitute.

   $150=\frac{10}{2}\cdot n=5n$

4. 4

   Solve for n.

   $n=\frac{150}{5}=30$

   $\boxed{30\ \text{days}}$

5. 5

   Check with a concrete progression. Any AP with $a_1+a_{30}=10$ works — for instance $a_1=1$ and $a_{30}=9$, so $d=\tfrac{9-1}{29}=\tfrac{8}{29}$. Its sum is $\tfrac{1+9}{2}\cdot 30=150$ m ✓. The individual daily distances are not determined, but the number of days is, because the sum formula only ever sees the pair $a_1+a_n$.