To find the equation for the nth term of the arithmetic sequence given by the numbers 3, 5, 7, and 9, we start by identifying the common difference of the sequence.
The sequence can be described as:
- Term 1 (n=1): 3
- Term 2 (n=2): 5
- Term 3 (n=3): 7
- Term 4 (n=4): 9
We can see that the difference between consecutive terms is constant. The common difference (d) can be calculated as follows:
- d = Term 2 – Term 1 = 5 – 3 = 2
- d = Term 3 – Term 2 = 7 – 5 = 2
- d = Term 4 – Term 3 = 9 – 7 = 2
Since the common difference is 2, we can use the formula for the nth term (Tn) of an arithmetic sequence, which is:
Tn = a + (n – 1) * d
Where:
- Tn is the nth term.
- a is the first term of the sequence (which is 3).
- n is the term number.
- d is the common difference (which is 2).
Plugging in the values:
Tn = 3 + (n – 1) * 2
We can simplify this further:
Tn = 3 + 2n – 2
Tn = 2n + 1
Thus, the equation for the nth term of the arithmetic sequence 3, 5, 7, 9 is:
Tn = 2n + 1
In conclusion, using the first term and the common difference, we have derived a straightforward equation that can be used to find any term in this particular arithmetic sequence.