Given the arithmetic series

Question

Given  the Arithmetic series  

8 +  11 + 14 +  17 + . . . + 77

What is the value of sum?

Answer 1

The way I think about summation of partial arithmetic series is to first find the average value A of the terms included, then multiply by the number of terms n. Because arithmetic series are evenly spaced, we can find the average for any segment by simply taking the average of the first and last terms:

A = (8 + 77)/2 = 42.5

To find the number of terms, we'll take the difference of the first and last terms and divide by the common difference between consecutive terms (thus finding the number of "steps" taken), then add 1 at the end:

n = (77 - 8)/3 + 1 = 23 + 1 = 24

Thus our final sum is A * n = 42.5 * 24 = 1020.

Answer 2

Please review the meaning of the terms "series" and "sequence".

a₁ = 8

any term a_n = a₁ + d(n-1)

77 = 8 + 3(n-1) [note that d=3]

69 = 3n - 3

72 = 3n

n = 24 [this means that 77 is the 24th term]

S = (n/2)(a₁ + a_n) [This is the sum of elements 1 to n]

S = (24/2)(8 + 77)

S = 12*85

S = 1020