Write an explicit and recursive formula for the sequence 4, 7, 10 …

Answer 1

Now because the question asks for a recursive formula, we first have to identify our base case, or in other words, our first term. If we denote our first term a₀, we see that it is 4. Because we want a recursive formula, we need to relate the next term in the sequence with the current term in the sequence. Let us begin by denoting the current term as a_n and the next term as a_n+1. We see that the next term is going to always be 3 more than our current term. For example, if our current term is 4, our next term is 3 more than that, i.e. it is 7. So we can create an equation using this relation, a_n+1 = 3 + a_n. This equation tells us that the next term is going to be 3 more than our current one.

So in short,

a₀ = 4

a_n+1 = 3 + a_n.

If we want an explicit formula, we want to create an equation that gives us the terms of our sequence. In other words, we want a sort of input/output kind of equation. Our input would be the position of the term and our output would be our term. In other words, let us say we want the term at the 3^rd position of the sequence. The input would be 3 and the output would be 7. Because the sequence increases by a common difference of 3, we can use our formula for arithmetic sequences, a_n = a₀ + d(n-1). "d" represents the difference between the second term and the first term, n represents the position number, a₀ represents our first term, and a_n represents the term at the position number n. We know the sequence increases by 3 every time and we also know that our first term is 4. If we plug everything in we have:

a_n = a₀ + d(n-1)

a_n = 4 + 3(n-1). Which is our formula.