An a.p has 20 terms and a common different of 3 determine the last term and of the a.p it's sum is (a)165 (b)650
Let the first term of the arithmetic progression be a and the last term be l.
The sum of an arithmetic progression is given by the formula:
Sum = n/2 * (2a + (n-1)d)
where n is the number of terms, a is the first term, and d is the common difference.
Given that there are 20 terms and the common difference is 3, we have:
n = 20
d = 3
The sum of the arithmetic progression is given as 165:
165 = 20/2 * (2a + 19 * 3)
165 = 10 * (2a + 57)
165 = 20a + 570
20a = 405
a = 20.25
Therefore, the first term of the arithmetic progression is 20.25.
Now, to find the last term, we use the formula for the nth term of an arithmetic progression:
l = a + (n-1)d
l = 20.25 + (20-1)3
l = 20.25 + 57
l = 77.25
Thus, the last term of the arithmetic progression is 77.25.
Similarly, for the sum to be 650:
650 = 20/2 * (2a + 19 * 3)
650 = 10 * (2a + 57)
650 = 20a + 570
20a = 80
a = 4
Therefore, the first term of the arithmetic progression is 4.
Using the formula for the last term:
l = 4 + (20-1)3
l = 4 + 57
l = 61
Thus, the last term of the arithmetic progression is 61.