Determine whether the following is a recursive or explicit. Then, find the first four terms of the following sequence.
a) a_n = 〖na〗_(n-1) where a_0 =5
b) a_n = a_(n-1) + 3a_(n-2) where a_0 = 1 and a_1 =2
c) a_n = 2^n + 5•3^n for n=0,1,2,3
if a_n depends on previous values of the sequence, then it is recursively defined.
To determine whether a sequence is recursive or explicit, we need to check if each term in the sequence is defined in terms of previous terms (recursive) or if each term can be directly calculated without any reference to previous terms (explicit).
a) For the sequence a_n = 〖na〗_(n-1) where a_0 = 5, each term is defined in terms of the previous term, so it is a recursive sequence. To find the first four terms, we can use the recursive formula:
a_0 = 5
a_1 = 1 * a_0 = 1 * 5 = 5
a_2 = 2 * a_1 = 2 * 5 = 10
a_3 = 3 * a_2 = 3 * 10 = 30
Therefore, the first four terms of this sequence are 5, 5, 10, 30.
b) For the sequence a_n = a_(n-1) + 3a_(n-2) where a_0 = 1 and a_1 = 2, each term is defined in terms of the two previous terms, so it is also a recursive sequence. We can use the recursive formula to find the first four terms:
a_0 = 1
a_1 = 2
a_2 = a_1 + 3a_0 = 2 + 3 * 1 = 5
a_3 = a_2 + 3a_1 = 5 + 3 * 2 = 11
Therefore, the first four terms of this sequence are 1, 2, 5, 11.
c) For the sequence a_n = 2^n + 5 * 3^n for n = 0, 1, 2, 3, each term can be directly calculated without reference to previous terms, so it is an explicit sequence. We can plug in the values of n to find the first four terms:
a_0 = 2^0 + 5 * 3^0 = 1 + 5 * 1 = 6
a_1 = 2^1 + 5 * 3^1 = 2 + 5 * 3 = 17
a_2 = 2^2 + 5 * 3^2 = 4 + 5 * 9 = 49
a_3 = 2^3 + 5 * 3^3 = 8 + 5 * 27 = 143
Therefore, the first four terms of this sequence are 6, 17, 49, 143.