Use the initial term and the recursive formula to find an explicit formula for the sequence an. Write your answer in simplest form. a1 = – 33 an = an–1+7
To find an explicit formula for the sequence, we need to examine how the terms relate to each other.
Let's write out the first few terms of the sequence to see if we can notice a pattern:
a1 = -33
a2 = a1 + 7 = -33 + 7 = -26
a3 = a2 + 7 = -26 + 7 = -19
a4 = a3 + 7 = -19 + 7 = -12
From this, we can see that each term is obtained by adding 7 to the previous term. In other words, we are increasing each term by 7 as we move through the sequence.
Now, let's examine this pattern in a more general way. We start with a1 = -33.
To get the next term, a2, we add 7 to a1: a2 = a1 + 7 = -33 + 7.
To get the next term, a3, we add 7 to a2: a3 = a2 + 7 = (-33 + 7) + 7.
To get the next term, a4, we add 7 to a3: a4 = a3 + 7 = ((-33 + 7) + 7) + 7.
We can continue this pattern and see that to get the next term, we add 7 to the previous term: an = an-1 + 7.
This recursive formula allows us to find each term of the sequence by adding 7 to the previous term.
Now, let's express this relationship using an explicit formula.
To find the explicit formula, we can use the fact that each term is obtained by adding 7 to the previous term. So, if we start with a1 = -33, we can express the n-th term as:
an = a1 + (n-1)d
where d is the common difference of 7.
Substituting the values, we have:
an = (-33) + (n-1)(7)
Simplifying this equation, we get:
an = -33 + 7(n-1)
Expanding, we have:
an = -33 + 7n - 7
Combining like terms, we get:
an = 7n - 40
Therefore, the explicit formula for the sequence an is an = 7n - 40.