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

##### 3 Answers

196

#### Explanation:

The sum to n terms of an Arithmetic sequence is given by:

# S_n = n/2 [ 2a + (n - 1 )d ]# where a , is the 1st term , d the common difference and n , the number of terms to be summed.

Here a = 1 , d = 2 and n = 14

#rArr S_14 = 14/2 [ (2xx1) + (13xx2) ] = 196 #

Solution reworked and found to be 196

#### Explanation:

Series:

Notice that each term is calculated by

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There is a trick for solving these

Consider the above table

So for an even count of values the sum is

We have

'~~~~~~~~~~~~~~~~~ Further comment ~~~~~~~~~~~~~~~~~

Suppose there had been an odd number of terms.

We could pair up our values as above but there would be a single unpaired value in the middle. In this case you would have:

The

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Development of solution formula being

#### Explanation:

Let a term in the sequence

Let the sum of this eries be

Then

Consequently

sum_(I=1ton)

But

but

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Suppose the series did not start at 1 but was say: 15 to 47

You could calculate the sum from 1 to 47 and then subtract from it the sum of 1 to 13. So you would have