A Diginacci sequence is created as follows. The first two terms are any positive whole numbers. Each of the remaining terms is the sum of the digits of the previous two terms For example, starting with 4 and 8 the Diginacci sequence is 4, 8, 13, 12, 7, 10, ... The calculations for this exmaple are 4 + 8 = 13, 8 + 1 + 3 = 12, 1 + 3 + 1 + 2 = 7, 1 + 2 + 7 = 10 a. Calculate the first 28 terms of the Diginacci sequence with starting terms 1 and 1, and then find the 2021st number in the sequence. b. Show that if both starting terms in a Diginacci sequence are ach less than one million, then its fourth and fifth terms are each less than 100. c. Show that if both starting terms in a Diginacci sequence are each less than 100, then it has a term after which all terms are at most 20. d. Show that if both starting terms in a Diginacci sequence are each less than 100, then it has a term after which all terms equal 18 or all terms are less than 18.
A Diginacci sequence is created as follows.
- The first two terms are any positive whole numbers.
- Each of the remaining terms is the sum of the digits of the previous two terms
For example, starting with 4 and 8 the Diginacci sequence is
4, 8, 13, 12, 7, 10, ...
The calculations for this exmaple are
4 + 8 = 13, 8 + 1 + 3 = 12, 1 + 3 + 1 + 2 = 7, 1 + 2 + 7 = 10
a. Calculate the first 28 terms of the Diginacci sequence with starting terms 1 and 1, and then find the 2021st number in the sequence.
b. Show that if both starting terms in a Diginacci sequence are ach less than one million, then its fourth and fifth terms are each less than 100.
c. Show that if both starting terms in a Diginacci sequence are each less than 100, then it has a term after which all terms are at most 20.
d. Show that if both starting terms in a Diginacci sequence are each less than 100, then it has a term after which all terms equal 18 or all terms are less than 18.
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