What is the value of the sum

Question

Given  the Arithmetic series  

13 +  16 + 19 +  22 + . . . + 67

What is the value of sum?

Answer 1

13 + 16 + 19 + 22 + . . . + 67

is 19 terms

a₁ = 13

a_n = 67



a_n = a₁ + d(n-1)

67 = 13 + 3(n-1)

67 = 13 + 3n - 3

57 = 3n

19 = n [67 is the 19th term]

S = (n/2)(a₁ + a_n)

S = (19/2)(13+67)

S = (19/2)(80)

S = 760

Answer 2

The sum of the first n terms of an arithmetic sequence is:

S = (n/2)(a₁ + a_n)

where n is the number of terms, a₁ is the first term, and a_n is the last term. In this case, a₁ = 13 and a_n = 67. So to find S we just need to know n, the number of terms. The explicit formula for the sequence is

a_n = a₁ + d·n where d = 3 = the common difference between terms.

a_n = a₁ + d·n

67 = 13 + 3n

54 = 3n

18 = n

Now just plug a₁ = 13, a_n = 67 and n = 18 into the equation for the sum S above and compute the answer.