Final answer:
The largest score that cannot be obtained solely with 3-point field goals and 7-point touchdowns is 5. This has been determined through base cases and using mathematical induction to show continuity of scoring beyond a sequence of six consecutive scores starting at 6.
Step-by-step explanation:
This problem is a classic example of mathematical induction and number theory, where the goal is to find the largest number of points which cannot be scored with combinations of 3-point field goals and 7-point touchdowns. Inductively, we can prove that all numbers starting from a certain point can be formed by 3 and 7. First, we need to establish the base cases, which are typically 3, 6, 9, and so on, and similarly 7, 14, 21, and so on. The smallest number that cannot be constructed from combining these two sequences is 1, 2, 4, and 5. The induction step shows that if we can make all numbers up to some number n, then by adding 3 (the smallest increment), we can make all numbers up to n+3.
The largest of these base cases that can't be formed is 5 because:
- 1, 2, and 4 cannot obviously be expressed as sums of 3s and 7s,
- The next number, 6, is just two 3s, and
- Five is the largest number that cannot be expressed as a sum of 3s and 7s before hitting 6, which we just showed can be expressed as such.
Therefore, once we reach a sequence of six consecutive numbers that can be scored (6, 7, 8, 9, 10, 11), where 6 and 7 are obvious as they are just two field goals and one touchdown respectively, we can keep adding field goals to these to get any higher number. Hence, the number 5 is the largest number that cannot be scored with 3-point field goals and 7-point touchdowns.