The 10th and 15th term of an A.P are -5 and -7½, respectively. What is the sum of the first 20th term
To find the sum of the first 20 terms of an arithmetic progression (AP), you need to know the first term, the last term, and the common difference.
Let's call the first term of the AP "a" and the common difference "d".
The 10th term is given as -5, so we can write the equation:
a + 9d = -5 ------(1)
The 15th term is given as -7.5, so we can write the equation:
a + 14d = -7.5 -------(2)
To find the values of "a" and "d", we can subtract equation (1) from equation (2):
(a + 14d) - (a + 9d) = -7.5 - (-5 )
a + 14d - a - 9d = -7.5 + 5
5d = -2.5
d = -2.5/5
d = -0.5
Substituting the value of d in equation (1):
a + 9(-0.5) = -5
a - 4.5 = -5
a = -5 + 4.5
a = -0.5
So, the first term (a) is -0.5, and the common difference (d) is -0.5.
Now we can find the sum of the first 20 terms using the formula for the sum of an AP:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the first n terms, and n is the number of terms.
Substituting the values:
S20 = (20/2)(2(-0.5) + (20-1)(-0.5))
S20 = 10(-1 + 19(-0.5))
S20 = 10(-1 + 19*-0.5)
S20 = 10(-1 - 9.5)
S20 = 10(-10.5)
S20 = -105
Therefore, the sum of the first 20 terms is -105.