Calculate S_58 for the arithmetic sequence {a_n}={5/6n+1/3}

Answer 1

To find the value of S_58, which represents the sum of the arithmetic sequence up to the 58th term, we can use the formula:

S_n = (n/2)(a_1 + a_n),

where S_n represents the sum of the arithmetic sequence up to the nth term, n represents the number of terms, a_1 represents the first term, and a_n represents the nth term.

First, let's find the value of a_1 and a_n.
Given that the arithmetic sequence is {a_n} = {5/6n + 1/3}. We can see that a_1 = 5/6(1) + 1/3 = 5/6 + 1/3 = 5/6 + 2/6 = 7/6.
Also, a_n = 5/6(n) + 1/3.

Using the formula S_n = (n/2)(a_1 + a_n):
S_58 = (58/2)(a_1 + a_58),
S_58 = 29(a_1 + a_58).

To calculate the value of a_58, we substitute n = 58 in the formula for a_n:

a_58 = 5/6(58) + 1/3 = 290/6 + 2/6 = 292/6 = 146/3.

Substituting the values, we have:
S_58 = 29(a_1 + a_58) = 29(7/6 + 146/3).

Now, we need to find the common denominator of 6 and 3, which is 6. Rewriting the fractions with the same denominator:
S_58 = 29(7/6 + 292/6),

Adding the fractions inside the parentheses:
S_58 = 29(299/6),

Now, we can multiply:
S_58 = (29 * 299)/6.

Finally, we can calculate the value of S_58:
S_58 = 8671/6.

Therefore, S_58 for the arithmetic sequence {a_n} = {5/6n + 1/3} is equal to 8671/6.