How can you represent the terms of a sequence explicitly? How can you represent them recursively?

depends a lot on the sequence.

To represent the terms of a sequence explicitly, you can use a formula or an equation to directly calculate each term of the sequence. This formula typically involves an expression in terms of the position, "n," of the term you want to find. For example, consider the arithmetic sequence: 2, 5, 8, 11, ...

To represent this sequence explicitly, you can use the formula:

an = a1 + (n - 1) * d

where "an" is the "n-th" term of the sequence, "a1" is the first term, "n" is the position or term number, and "d" is the common difference between terms. In this case, the first term (a1) is 2 and the common difference (d) is 3. So, for example, to find the 5th term (n = 5), you can substitute the values into the formula:

a5 = 2 + (5 - 1) * 3
= 2 + 4 * 3
= 2 + 12
= 14

Thus, the 5th term of the sequence is 14.

On the other hand, to represent the terms of a sequence recursively, you define each term in relation to one or more previous terms. This means that to find a term, you need to know the value(s) of one or more preceding terms in the sequence. Recursion is often defined using a recursive formula or an equation.

For example, consider the Fibonacci sequence: 0, 1, 1, 2, 3, 5, ...

To represent this sequence recursively, you can use the following formulas:

F(0) = 0, (base case 1: first term)
F(1) = 1, (base case 2: second term)
F(n) = F(n-1) + F(n-2) (recursive formula: for n>1)

Here, F(n) represents the "n-th" term of the sequence. The base cases define the first two terms directly, and the recursive formula defines each subsequent term as the sum of the two preceding terms. Using this recursion, you can find any term in the sequence by recursively evaluating the formula.

For example, to find the 6th term (n = 6), you can use the recursion:

F(6) = F(5) + F(4)
= (F(4) + F(3)) + (F(3) + F(2))
= [(F(3) + F(2)) + (F(2) + F(1))] + [(F(2) + F(1)) + F(1)]
= [(F(2) + F(1)) + F(1)] + [F(1) + 1]
= [F(1) + 1] + 1
= 1 + 1
= 2

Thus, the 6th term of the Fibonacci sequence is 2.