The fifth term of a Ap is -1 and the sum of the first twenty term is -240. Find the third term
a+4d = -1
20/2 (2a+19d) = -240
Solve and get a=7, d = -2
Now find a3 = a+2d
Let's assume that the first term of the arithmetic progression (AP) is 'a' and the common difference is 'd'.
We are given that the fifth term of the AP is -1, so using the formula for the nth term of an AP, we have:
a + (5-1)d = -1
a + 4d = -1 ---(1)
We are also given that the sum of the first 20 terms of the AP is -240. The formula for the sum of the first n terms of an AP is given by:
Sn = (n/2)(2a + (n-1)d)
Plugging in the values, we have:
20/2 * (2a + 19d) = -240
10 * (2a + 19d) = -240
2a + 19d = -24 ---(2)
We have two equations (1) and (2) with two unknowns (a and d). We can solve these equations simultaneously to find the values of a and d.
Equation (1): a + 4d = -1
Equation (2): 2a + 19d = -24
We can solve these equations using any method such as substitution or elimination.
Let's solve using elimination method:
Multiply equation (1) by 2, we have:
2a + 8d = -2 ---(3)
Now, subtract equation (3) from equation (2):
(2a + 19d) - (2a + 8d) = -24 - (-2)
11d = -22
d = -2
Substituting d = -2 into equation (1):
a + 4(-2) = -1
a - 8 = -1
a = 7
Now that we have found the values of a and d, we can find the third term of the AP using the formula for the nth term of an AP:
a3 = a + (3-1)d
a3 = 7 + 2
a3 = 9
Therefore, the third term of the AP is 9.