1. To find two starting terms for a Diginacci sequence such that the 2021st term is 11, we can work backward from the 2021st term to determine the preceding terms.
Let's denote the first term as a and the second term as b. We need to find the values of a and b that will result in the 2021st term being 11.
Working backward, we know that the 2021st term (which is 11) is the sum of the digits of the previous two terms. Let's denote the (2020)th term as x and the (2019)th term as y. Therefore, we have:
11 = x + y
2. To find a suitable pair of x and y, we can iterate through possible values. Since the first two terms can be any positive whole numbers, we have some flexibility in choosing them.
For example, let's try a = 1 and b = 1:
Term 1: 1
Term 2: 1
Term 3: 2
Term 4: 1
Term 5: 3
Term 6: 4
Term 7: 7
Term 8: 11
The 8th term is 11, which matches our target. Therefore, if we start with a = 1 and b = 1, the 2021st term will indeed be 11.
To find a Diginacci sequence that has no term equal to 11, we need to ensure that none of the subsequent terms sum up to 11.
One way to achieve this is to start with two terms that are relatively large and have no digits summing up to 11. Let's try a = 99 and b = 100:
Term 1: 99
Term 2: 100
Term 3: 19 (9 + 9)
Term 4: 10 (1 + 0)
Term 5: 1
Term 6: 1
Term 7: 2
Term 8: 3
Term 9: 5
Term 10: 8
Term 11: 13
Term 12: 12
Term 13: 7
Term 14: 10
...
By starting with a = 99 and b = 100, the Diginacci sequence continues without reaching the value 11. The terms keep changing without summing up to 11, as seen in the subsequent terms above.
Thus, this particular Diginacci sequence starting with a = 99 and b = 100 does not have any term equal to 11.