Use the arithmetic sequence nth term formula to solve the following problem:
The first, second, and the nth terms of an arithmetic sequence are 2, 6, and 58 respectively:
find the value of n
and For that value of n, what is the exact value of the sum of n terms.
the common difference is ... 6 - 2
the change from 6 to 58 is ... 58 - 6
the number of terms from 6 to 58 is ... (58 - 6) / (6 - 2)
... add in the 1st two terms to find n
sum = (58 + 2) * (n / 2)
To solve this problem, we can use the formula for the nth term (a_n) of an arithmetic sequence:
a_n = a_1 + (n-1)d
where:
a_n is the nth term,
a_1 is the first term, and
d is the common difference between consecutive terms.
Given that the first term (a_1) is 2, and the second term (a_2) is 6, we can find the common difference (d) using the formula:
a_2 = a_1 + d
6 = 2 + d
d = 6 - 2
d = 4
Now, we can substitute the values for a_1, d, and a_n into the formula and solve for n:
a_n = a_1 + (n-1)d
58 = 2 + (n-1)4
Simplifying the equation:
58 = 2 + 4n - 4
58 = 4n - 2
60 = 4n
n = 60/4
n = 15
Therefore, the value of n is 15.
Now that we know the value of n, we can find the sum of n terms using the formula:
S_n = (n/2)(a_1 + a_n)
Substituting the values we have:
S_15 = (15/2)(2 + 58)
S_15 = (15/2)(60)
S_15 = 15 * 30
S_15 = 450
So, the exact value of the sum of the 15 terms is 450.