Write a recursive definition for the sequence 11, 8, 5, 2,…
To write a recursive definition for the sequence 11, 8, 5, 2, ... we need to identify how each term is related to the previous term.
In this sequence, each term is obtained by subtracting 3 from the previous term:
a1 = 11
a2 = a1 - 3 = 8
a3 = a2 - 3 = 5
a4 = a3 - 3 = 2
...
Thus, we can write the recursive definition:
an = an-1 - 3 for n >= 2
where a1 = 11 is the initial term of the sequence.
To write a recursive definition for a sequence, we need to express each term in the sequence using the ones that come before it. In this case, the sequence is 11, 8, 5, 2, and so on.
To define the sequence recursively, we can say that the first term is 11: a₁ = 11.
Then, for each term after the first one, we can say that it is obtained by subtracting 3 from the previous term: aₙ = aₙ₋₁ - 3.
So, in a recursive definition, the sequence can be defined as:
a₁ = 11
aₙ = aₙ₋₁ - 3, for n > 1.
Using this recursive definition, we can generate the terms of the sequence by substituting the values of n into the definition. For example, to find the 5th term, we can substitute n = 5 into the definition:
a₅ = a₅₋₁ - 3 = a₄ - 3 = (a₃ - 3) - 3 = ((a₂ - 3) - 3) - 3 = (((a₁ - 3) - 3) - 3) = 11 - 3 - 3 - 3 = 2.
Therefore, the 5th term of the sequence is 2.
To find a recursive definition for the given sequence 11, 8, 5, 2, ..., we can start by identifying the pattern between consecutive terms.
From the first term to the second term, we can observe that the value decreases by 3. Similarly, from the second term to the third term, the value again decreases by 3. This pattern continues throughout the sequence.
Based on this observation, we can define a recursive formula to generate the terms of the sequence:
Step 1: Define the first term of the sequence:
a(1) = 11
Step 2: Define the recursive formula to generate subsequent terms:
a(n) = a(n-1) - 3
This defines the recursive definition for the given sequence. It states that each term in the sequence can be obtained by subtracting 3 from the previous term.