The fifth term of an arithmetic sequence is 23 and the 12th term is 72. Determine the first three terms of the sequence and the nth term
Let's denote the first term of the arithmetic sequence as 'a' and the common difference as 'd'.
The fifth term of the sequence can be expressed as:
a + 4d = 23 (equation 1)
The twelfth term of the sequence can be expressed as:
a + 11d = 72 (equation 2)
Now, let's solve these two equations to find the values of 'a' and 'd'.
Multiplying equation 1 by 11 and equation 2 by 4, we get:
11(a + 4d) = 11(23) -> 11a +44d = 253 (equation 3)
4(a + 11d) = 4(72) -> 4a + 44d = 288 (equation 4)
Subtracting equation 3 from equation 4 to eliminate 'd', we get:
4a + 44d - 11a - 44d = 288 - 253
-7a = 35
a = -5
Now, substitute the value of 'a' in equation 1 to solve for 'd':
-5 + 4d = 23
4d = 28
d = 7
Therefore, the first term of the sequence (a) is -5 and the common difference (d) is 7.
The first three terms of the sequence are:
Term 1: a = -5
Term 2: a + d = -5 + 7 = 2
Term 3: a + 2d = -5 + 2(7) = 9
The nth term of the sequence can be represented as:
a + (n-1)d
So, the nth term is -5 + (n-1)(7) = -5 + 7n - 7 = 7n - 12.