Use Gausss approach to find the following sum do not use for

Use Gauss\'s approach to find the following sum (do not use formulas): 8 + 14 + 20 + 26+ ... + 86. The sum of the sequence is .

Solution

The sequence given is: 8+14+20+............+86

Now to calculate the sum of this sequence using Gauss approach we use the following steps:

Now, 8+86=94;14+80=94;20+74=94 and so on..(basically we are trying to show that the sum of extreme terms.ie.the sum of 1st and last term,2nd and the second last term and so on, are the same in a sequence with common difference)

Now,we have to find the number of terms in the sequence.

8=8+0*6 (We are using 6 here as the common difference between the terms of the sequence is 6)

14=8+1*6

20=8+2*6............

86=8+13*6

Hence,the no of terms(n) is 14......Therefore no of pairs is : n/2=14/2=7....This indicates that we have 7 pairs of numbers whose sum is 94.

The sum of the sequence is= 94*7=658