Determine the general rule(Tn) in the following pattern, and show all your calculations
2,6,10,14
The general rule is Tn = 4n - 2.
Calculations:
- T1 = 4(1) - 2 = 2
- T2 = 4(2) - 2 = 6
- T3 = 4(3) - 2 = 10
- T4 = 4(4) - 2 = 14
To determine the general rule (Tn) for the given pattern 2, 6, 10, 14, we need to examine the differences between consecutive terms in the sequence.
The first difference (d1) between the second and first terms is:
6 - 2 = 4
The second difference (d2) between the third and second terms is:
10 - 6 = 4
Similarly, the third difference (d3) between the fourth and third terms is:
14 - 10 = 4
Since the differences between the consecutive terms are all the same (4), we can conclude that the general rule for this pattern is an arithmetic sequence.
Now, let's find the formula for the nth term (Tn) in the arithmetic sequence.
We know that the first term (T1) is 2, and the common difference (d) is 4.
The formula for the nth term of an arithmetic sequence (Tn) is given by:
Tn = T1 + (n - 1) * d
Substituting the given values:
Tn = 2 + (n - 1) * 4
Simplifying:
Tn = 2 + 4n - 4
Tn = 4n - 2
Therefore, the general rule for the given pattern is Tn = 4n - 2.
To verify the calculations:
For n = 1:
T1 = 4(1) - 2 = 2
For n = 2:
T2 = 4(2) - 2 = 6
For n = 3:
T3 = 4(3) - 2 = 10
For n = 4:
T4 = 4(4) - 2 = 14
As we can see, these calculations match the given sequence correctly.