what is the 15th term of an arithmetic sequence in which the third term is 1985 and the eighth term in 2015?
a + d(n-1)
a + d (2) = 1985
a + d(7) = 2015
----------------subtract
0 - 5 d = - 30
d = 6
a + 12 = 1985
a = 1973
1973 + 6(14) = 2057
To find the 15th term of an arithmetic sequence, we need to know the common difference (d). The common difference is the constant value that is added or subtracted to get to the next term.
Given that the third term is 1985 and the eighth term is 2015, we can use this information to find the common difference.
Step 1: Find the value of the fifth term:
Since the common difference is added each time, the fifth term is two positions away from the third term. Therefore, we can find it by adding the common difference twice to the third term:
1985 + d + d = 1985 + 2d
Step 2: Find the value of the eighth term:
Since the common difference is added each time, the eighth term is five positions away from the third term. Therefore, we can find it by adding the common difference five times to the third term:
1985 + 5d
Step 3: Set up an equation:
We know that the eighth term is 2015, so we can write the following equation:
1985 + 5d = 2015
Step 4: Solve the equation for d:
Subtract 1985 from both sides of the equation:
5d = 2015 - 1985
5d = 30
Divide both sides of the equation by 5:
d = 30 / 5
d = 6
Step 5: Calculate the 15th term:
Now that we know the common difference (d = 6), we can find the 15th term. The 15th term is 12 positions away from the third term, so we need to add the common difference 12 times:
1985 + (12 * 6) = 1985 + 72 = 2057
Therefore, the 15th term of the arithmetic sequence is 2057.