the 10th term of an arithmetic series is 34, and the sum of the first 20 terms is 710, determine the 25th term.
I will assume that you know the formulas for term(n) and sum(n) of an AS
10 term is 34 ---> a+9d = 34
sum of 20 terms is 710 ---> (20/2)[2a + 19d] = 710
2a + 19d = 71
double the first equation, then subract
2a + 18d = 68
d = 3
back into 1st ---> a + 27 = 34
a = 7
term(25) = a+24d = 7 + 72 = 79
To determine the 25th term of an arithmetic series, we need to use the given information about the 10th term and the sum of the first 20 terms.
Step 1: Find the common difference (d)
In an arithmetic series, each term is obtained by adding a constant value called the common difference (d) to the previous term. We can find the common difference using the 10th term (a10) and the formula for the nth term (an) of an arithmetic series:
an = a1 + (n - 1)d
Given that a10 = 34, we can substitute these values into the formula and solve for d:
34 = a1 + (10 - 1)d
Step 2: Find the first term (a1)
To find the first term (a1), we can substitute the known values into the same formula, using the 10th term and the common difference (d):
34 = a1 + (10 - 1)d
Step 3: Find the sum of the first 20 terms (S20)
The sum of the first 20 terms (S20) can be calculated using the formula for the sum of an arithmetic series:
S20 = (n/2)(a1 + a20)
Given that S20 = 710, we can substitute the values into the formula and solve for a1:
710 = (20/2)(a1 + a20)
Step 4: Find the 25th term (a25)
Now that we have found the first term (a1) and the common difference (d), we can find the 25th term (a25) using the formula for the nth term:
a25 = a1 + (25 - 1)d
Substitute the values we found into the formula:
a25 = a1 + (25 - 1)d
By solving these equations step-by-step, we can determine the 25th term of the arithmetic series.