How do you find the sum of the arithmetic series 1 + 3 + 5 + ... + 27?
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
Notice that each term is calculated by
There is a trick for solving these
Consider the above table
So for an even count of values the sum is
'~~~~~~~~~~~~~~~~~ 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:
Development of solution formula being
Let a term in the sequence
Let the sum of this eries be
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