find the sum of the 20th term a=6 and r=2/3
To find the sum of the 20th term of an arithmetic series, we can use the formula:
Sn = (n/2)(2a + (n-1)d)
Where Sn is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
In this case, a = 6, r = 2/3, and we need to find the sum of the 20th term.
First, we need to find the 20th term. We can use the formula for the nth term of an arithmetic series:
an = a + (n-1)d
In this case, d is the common ratio, so we substitute d = r:
a20 = 6 + (20-1)(2/3)
= 6 + (19)(2/3)
To simplify, we can multiply 19 by 2 and then divide by 3:
a20 = 6 + (38/3)
= 18/3 + 38/3
= 56/3
Now that we have found the value of the 20th term (a20 = 56/3), we can substitute it back into the formula for the sum of the first n terms:
Sn = (n/2)(2a + (n-1)d)
Here, n = 20, a = 6, and d = r = 2/3:
S20 = (20/2)(2(6) + (20-1)(2/3))
= 10(12 + (19)(2/3))
To simplify, we can multiply 19 by 2 and then divide by 3:
S20 = 10(12 + 38/3)
= 10(36/3 + 38/3)
= 10(74/3)
= (740/3)
Therefore, the sum of the 20th term is 740/3.