Question

How do you find the sum of the arithmetic series 1 + 3 + 5 + ... + 27?

Answer 1

sum = 196

Explanation:

the sum to n terms of an arithmetic series, when the first and last terms are known is

`S_(n)` =
`(n)/(2)` ( first term + last term)

we require to find the number of terms in the series

the nth term of an arithmetic series is

`a_(n)` = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = 1 and d = a₂ - a₁ = 3 - 1 = 2 , then

1 + 2(n - 1) = 27 ( subtract 1 from both sides )

2(n - 1) = 26 ( divide both sides by 2 )

n - 1 = 13 ( add 1 to both sides )

n = 14 ← number of terms in the series

Then

S₁₄ =
`(14)/(2)` (1 + 27) = 7 × 28 = 196