The 3rd and 12th terms of an arithmetic progression are 12 and -24 respectively determine the 50th term
To find the 50th term of an arithmetic progression, we need to first determine the common difference (d) between terms.
Given that the 3rd term of the arithmetic progression is 12 and the 12th term is -24, we can use these values to find the common difference.
The formula for the nth term of an arithmetic progression is given by:
a_n = a_1 + (n-1)d
Where:
a_n is the nth term
a_1 is the first term
d is the common difference
n is the term number
Using the information given:
a_3 = a_1 + (3-1)d = 12 --- (1)
a_12 = a_1 + (12-1)d = -24 --- (2)
Now, we can solve equations (1) and (2) simultaneously to find the values of a_1 and d.
Subtracting equation (1) from equation (2), we get:
(a_1 + 11d) - (a_1 + d) = -24 - 12
10d = -36
d = -3.6
Now that we have the value of d, we substitute it back into either equation (1) or (2) to find a_1.
Substituting d = -3.6 into equation (1), we get:
a_1 + (3-1)(-3.6) = 12
a_1 - 7.2 = 12
a_1 = 12 + 7.2
a_1 = 19.2
So, the first term (a_1) is 19.2 and the common difference (d) is -3.6.
Now that we know the first term and the common difference, we can use the formula for the nth term to find the 50th term (a_50) of the arithmetic progression.
a_50 = a_1 + (50-1)d
a_50 = 19.2 + 49(-3.6)
a_50 = 19.2 - 176.4
a_50 = -157.2
Therefore, the 50th term of the arithmetic progression is -157.2.
To determine the 50th term of the arithmetic progression, we can first find the common difference.
The common difference (d) can be found by subtracting the 3rd term from the 12th term.
12th term - 3rd term = -24 - 12 = -36
Now that we know the common difference, we can use it to find the formula for the nth term of an arithmetic progression, which is given by:
a + (n - 1) * d
where 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
We are given the 3rd term, which is 12. Plugging in this value, as well as the common difference (-36), into the formula, we can solve for the first term (a).
12 = a + (3 - 1) * (-36)
12 = a - 72
a = 12 + 72
a = 84
Now that we know the first term (a = 84) and the common difference (d = -36), we can find the 50th term using the same formula:
50th term = a + (50 - 1) * d
50th term = 84 + 49 * (-36)
50th term = 84 - 1764
50th term = -1680
Therefore, the 50th term of the arithmetic progression is -1680.
clearly, a_12 - a_3 = 9d = -36, so d = -4
a_50 = a_12 + 38d = -24 + 38(-4) = ____