In each of 8-10 (a) suppose a sequence of the L17r. where t 0, satisfies the given recurre relation (but not ecessarily the initial conditions), and fine possible values of (b) suppose a sequence satisfies theg initial conditions as well as the recurrence relation, and line explicit formula for the sequence. 8. ag = 2a-1+3a-2, for all integers k 2 aɔ = 1, a¡ = 2

Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .

A young pair of rabbits (one of each gender) is placed on an island. A pair of rabbits does not breed until they are 3 months old. After they are 3 months old, each pair of rabbits produces another pair each month. Find a recurrence relation for the number of pairs of rabbits on the island after n months, assuming that rabbits never die. Find the number of rabbits after 7 months.

Let an be the number of ways there are to toss a coin until it ends (for the first time) on double heads. Let bn be the number of ways there are to toss a coin such that double heads never occurs. Find the recurrence relations with initial conditions for both an and bn.