To find the formula for the nth term of an arithmetic sequence, we need to determine its first term and the common difference. An arithmetic sequence is one where each term after the first is generated by adding a constant (the common difference) to the previous term.
Given:
- a20 = 240
- a15 = 170
The general formula for the nth term of an arithmetic sequence can be expressed as:
an = a1 + (n – 1) * d
where:
- a1 is the first term,
- d is the common difference,
- n is the term number.
From the information provided, we can establish the following equations:
- For the 20th term (n=20):
- a20 = a1 + (20 – 1) * d = 240
This simplifies to:
a1 + 19d = 240
- For the 15th term (n=15):
- a15 = a1 + (15 – 1) * d = 170
This simplifies to:
a1 + 14d = 170
Now we have a system of two equations:
- 1) a1 + 19d = 240
- 2) a1 + 14d = 170
To solve this system, we can eliminate a1 by subtracting the second equation from the first:
(a1 + 19d) – (a1 + 14d) = 240 – 170
This results in:
5d = 70
Now, we can solve for d:
d = 70 / 5 = 14
Having found d, we can substitute it back into one of the earlier equations to find a1. Let’s use the second equation:
a1 + 14 * 14 = 170
This simplifies to:
a1 + 196 = 170
Thus:
a1 = 170 – 196 = -26
Now that we have both values, we can write the formula for the nth term:
an = -26 + (n – 1) * 14
This simplifies further to:
an = -26 + 14n – 14
Resulting in:
an = 14n – 40
Therefore, the rule for the nth term of the arithmetic sequence is:
an = 14n - 40