How do I find the sum using formulas and expressed in summat

Solution

3 + 11 + 19 +...+ 731

common difference of terms = (11 - 3) = (19 - 11) = 8

n^th term = a + (n -1) d (here a = 3 , d = 8)

==> T_n = 3 + (n -1) (8)

==> T_n = 3 + 8n -8

==> T_n = 8n - 5

Given last term = 731

==> 8n - 5 = 731

==> 8n = 731 + 5 = 736

==> n = 736/8 = 92

Hence total number of terms in the series = 92

Hence 3 + 11 + 19 +...+ 731 = _{[ n = 1 to 92]} T_n

==> _{[ n = 1 to 92]} (8n - 5)

==> 8 _{[n = 1 to 92]} n - 5(92) (since -5 occurs 92 times)

we know that sum of first n natural numbers = n(n +1)/2

==> Sum of first 92 natural numbers = 92(92 + 1)/2

==> 8 [92(92 + 1)/2] - 5(92)

==> 4(92*93) - 5(92)

==> 34224 - 460

==> 33764

Hence 3 + 11 + 19 +...+ 731 = 33764