To help pass time on a long bus ride, \(35\) math teachers created a sequence of numbers, with each teacher saying one term in the sequence. The first teacher said the number \(2\), the second teacher said the number \(8\), and every teacher after that said the sum of the two previous terms. Thus,
the third teacher said the sum of the first and second terms, which is \(2+8=10\), and
the fourth teacher said the sum of the second and third terms, which is \(8+10=18\).
Once the final teacher said their number, the 25th teacher announced they had made a mistake and their number should have been one more than what they had said. How much larger should the final teacher’s number have been?
Solution 1
We will write out the sequence of numbers the teachers actually said, and then the sequence of numbers they should have said, and then find the difference between the last term in each sequence.
Here are the first \(24\) numbers that the teachers said:
\(2\), \(8\), \(10\), \(18\), \(28\), \(46\), \(74\), \(120\), \(194\), \(314\), \(508\), \(822\), \(1330\), \(2152\), \(3482\), \(5634\), \(9116\), \(14\,750\), \(23\,866\), \(38\,616\), \(62\,482\), \(101\,098\), \(163\,580\), \(264\,678\)
Here are the correct 25th to 35th numbers that the teachers should have said:
\(428\,258\), \(692\,936\), \(1\,121\,194\), \(1\,814\,130\), \(2\,935\,324\), \(4\,749\,454\), \(7\,684\,778\), \(12\,434\,232\), \(20\,119\,010\), \(32\,553\,242\), \(52\,672\,252\)
Here are the 25th to 35th numbers that the teachers actually said:
\(428\,257\), \(692\,935\), \(1\,121\,192\), \(1\,814\,127\), \(2\,935\,319\), \(4\,749\,446\), \(7\,684\,765\), \(12\,434\,211\), \(20\,118\,976\), \(32\,553\,187\), \(52\,672\,163\)
The difference between the correct and incorrect 35th number is \(52\,672\,252-52\,672\,163=89\). Therefore, the 35th number was off by \(89\), and so the final teacher’s number should have been \(89\) larger than the number they had said.
Solution 2
In this solution we will solve the problem without actually calculating all the terms in the sequence.
We know the 25th term is off by \(1\). Therefore, the next terms will be as follows.
The 26th term will also be off by \(1\) since it equals the sum of the \(24^\textrm{th}\) term (which is unchanged) and the 25th term (which is off by \(1\)).
The 27th term will be off by \(2\) since it is the sum of the 25th term (which is off by \(1\)) and the 26th term (which is off by \(1\)).
The \(28^\textrm{th}\) term will be off by \(3\) since it is the sum of the 26th term (which is off by \(1\)) and the 27th term (which is off by \(2\)).
This pattern will continue on, so we can summarize it in a table.
| Term Number | Amount Below the Correct Value |
|---|---|
| 24 | 0 |
| 25 | 1 |
| 26 | 1 |
| 27 | 2 |
| 28 | 3 |
| 29 | 5 |
| 30 | 8 |
| 31 | 13 |
| 32 | 21 |
| 33 | 34 |
| 34 | 55 |
| 35 | 89 |
Therefore, the 35th term was off by \(89\), and so the final teacher’s number should have been \(89\) larger than the number they had said.
Notice that the terms in the right column of the table follow the same rule as the original question. That is, each term is the sum of the previous two terms.
For Further Thought: The last \(11\) numbers in the right column of the table are the first \(11\) numbers of a famous sequence known as the Fibonacci Sequence. You may wish to investigate the Fibonacci Sequence further.