The following sequence of numbers forms a quadratic sequence:
–3, –2, –3, –6, –11, …...
The first differences of the sequence also form a sequence. Determine an expression for general term of the first differences
-3
....... +1
-2 .............. -2
....... -1
-3 .............. -2
....... -3
-6 ............ -2
....... - 5
-11
To determine the general term of the first differences of the given sequence, we need to find the difference between consecutive terms. Let's denote the given sequence as an, where n represents the position of each term.
Given sequence: –3, –2, –3, –6, –11
To find the first differences, we subtract the previous term from the current term:
–2 – (–3) = 1
–3 – (–2) = –1
–6 – (–3) = –3
–11 – (–6) = –5
The first differences are: 1, –1, –3, –5
Notice that the difference between the terms of the sequence of first differences is not constant. This means that the sequence does not have a linear pattern.
However, we can see that the sequence of first differences has a quadratic pattern. We can write the general term of a quadratic sequence as:
an = an-1 + dn
Here, dn represents the difference between the terms of the sequence of first differences.
Based on the given first differences sequence: 1, –1, –3, –5
We can write the general term of the first differences as:
an = an-1 + (–2n + 3)
Therefore, the general term of the first differences of the given sequence is:
an = an-1 + (–2n + 3)
To determine the general term of the first differences, we need to find the differences between consecutive terms in the given sequence.
The given sequence is:
–3, –2, –3, –6, –11,...
To find the differences between consecutive terms, we subtract the previous term from the current term:
–2 - (-3) = 1,
–3 - (-2) = -1,
–6 - (-3) = -3,
–11 - (-6) = -5.
The first differences sequence is:
1, -1, -3, -5,...
To express the general term of the first differences sequence, we need to find a pattern or relationship between the terms.
Looking at the first differences, we can observe that each term decreases by 2 compared to the previous term. Therefore, we can write the general term of the first differences sequence as:
an = a1 + (n - 1) * d,
where:
an is the nth term of the first differences sequence,
a1 is the first term of the first differences sequence,
n is the position of the term we want to find,
and d is the common difference.
In this case, the first term (a1) is 1, and the common difference (d) is -2.
So, the general term of the first differences sequence is:
an = 1 + (n - 1) * (-2).
Therefore, the expression for the general term of the first differences is:
an = 1 - 2n + 2.