The 10th term of an Arithmetic Progressions is -27 and the 5th term is -12, what is the 18th term? Find also the sum of its 25 terms.
a + 4d = -12
a + 9d = -27
clearly, d = -3, so use that to find a, and then find
T18 = T10 + 8d = -27 - 24 = -51
S25 = 25/2 (2a + 24d) = _____
To find the 10th term of an arithmetic progression (AP), we need to know the first term (a) and the common difference (d). However, in this case, we are given the 5th term and the 10th term.
Let's first find the common difference (d).
The 10th term is given as -27. So, we can write:
a + 9d = -27 --------(1) [Since the 10th term = a + 9d]
The 5th term is given as -12. So, we can write:
a + 4d = -12 --------(2) [Since the 5th term = a + 4d]
Now, we have a system of two equations with two variables (a and d). We can solve this system of equations to find the values of a and d.
To solve the equations, we subtract equation (2) from equation (1) to eliminate 'a':
(a + 9d) - (a + 4d) = -27 - (-12)
5d = -15
d = -15/5
d = -3
Now that we have found the common difference (d), we can substitute it back into equation (2) to solve for 'a':
a + 4(-3) = -12
a - 12 = -12
a = 0
So, the first term (a) is 0 and the common difference (d) is -3.
Now, let's find the 18th term.
The formula to find the nth term of an arithmetic progression is:
an = a + (n - 1)d
Substituting the values into the formula:
a18 = 0 + (18 - 1)(-3)
a18 = 0 + 17(-3)
a18 = 0 - 51
a18 = -51
Therefore, the 18th term of the arithmetic progression is -51.
To find the sum of the 25 terms, we can use the formula for the sum of an arithmetic progression:
Sn = (n/2)(2a + (n - 1)d)
Substituting the values into the formula:
S25 = (25/2)(2(0) + (25 - 1)(-3))
S25 = (25/2)(0 + 24(-3))
S25 = (25/2)(0 - 72)
S25 = (25/2)(-72)
S25 = -900
Therefore, the sum of the 25 terms of the arithmetic progression is -900.