Find the rule for the sequence:
4.2, 2.6, 1, -.06, -2.2
put it in nth term
If you take the difference between consecutive terms, you will find that the difference is constant at -1.6.
So it is an arithmetic sequence.
T(n)=K-1.6n
T(1)=4.2
so 4.2=K-1.6*1 => K=4.2+1.6=5.8, or
T(n)=5.8-1.6n
Check: third term
T(3)=5.8-1.6*3=1.0 OK.
To find the rule for the sequence, let's examine the differences between consecutive terms:
2.6 - 4.2 = -1.6
1 - 2.6 = -1.6
-0.06 - 1 = -1.06
-2.2 - (-0.06) = -2.14
The differences seem to be constant at -1.6. This indicates that the sequence follows a linear pattern.
To confirm this, let's perform the next difference:
-1.6 - (-2.14) = 0.54
The difference is not exactly the same as the previous ones, indicating that the sequence might not be strictly linear.
However, we can still approximate a rule for this sequence by assuming a linear relationship.
If we assume a linear relationship for this sequence, the generic formula can be written as:
an = a1 + (n-1)d
where an represents the nth term of the sequence, a1 is the first term, n is the position of the term, and d is the common difference between consecutive terms.
Using this formula, we can plug in values from the sequence:
a1 = 4.2
d = -1.6 (approximated common difference)
Now, let's calculate the values for n = 2, 3, 4, and 5:
a2 = 4.2 + (2-1)(-1.6) = 2.6
a3 = 4.2 + (3-1)(-1.6) = 1
a4 = 4.2 + (4-1)(-1.6) = -0.4
a5 = 4.2 + (5-1)(-1.6) = -2.2
These values match the original terms of the sequence, confirming that the rule for the sequence is approximated by an = 4.2 + (n-1)(-1.6).