an arithmetic progression has 20 terms. the sum of all the even terms is 350. given that the first term is 5, find the common difference and the eleventh term.
To find the common difference and the eleventh term of an arithmetic progression, follow the steps below:
Step 1: Find the sum of all the even terms.
Given that the sum of all the even terms is 350, we can use the formula for the sum of an arithmetic sequence:
Sum = (n/2) * (2a + (n - 1) * d)
n = number of terms = 20
a = first term = 5
Sum = 350
Substituting these values into the formula, we can solve for the sum of the even terms:
350 = (20/2) * (2(5) + (20 - 1) * d)
350 = 10 * (10 + 19d)
35 = 10 + 19d
19d = 35 - 10
19d = 25
d = 25/19
Hence, we have found the value of the common difference: d = 25/19.
Step 2: Find the eleventh term.
To find the eleventh term, we can use the formula for the nth term of an arithmetic sequence:
An = a + (n - 1) * d
n = 11
a = 5
d = 25/19
Substituting these values into the formula, we can solve for the eleventh term:
A11 = 5 + (11 - 1) * (25/19)
A11 = 5 + 10 * (25/19)
A11 = 5 + (250/19)
A11 = (5 * 19 + 250) / 19
A11 = (95 + 250) / 19
A11 = 345/19
Hence, the eleventh term of the arithmetic progression is 345/19.