Use mathematical induction to prove the formula for every positive integer n. Please show all your work.

9+14+19+80+....+(5n+4)= [n/2] (5n+13)

Show that this is true for  n = 1

(1/2)(5[1] + 13]  =  (1/2)(18)  =  9

Assume that it is true for n =  k.....that is....

9 + 14 + 19 + 80 +........+ (5k + 4)  = (k/2) (5k + 13)      (1)

Prove that it is true for n = k + 1

That is

9 + 14 + 19 + 80 +........+  (5(k + 1) + 4 )  = [ (k + 1)/2 ] [5(k + 1) + 13 ]

Add   (5(k + 1) + 4 )    to both sides of (1)

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  =   (k /2)(5k + 13) + (5(k + 1) + 4)

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  =   (k /2)(5k + 13) + (5k + 9)

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  =    [ (k)(5k + 13) + (10k + 18 ] /2

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  =    [ 5k^2 + 13k + 10k + 18] /2

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  =    [ 5k^2 + 23k + 18] /2

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  =    [ (k + 1) (5k + 18)] /2

9 + 14 + 19 + 80 +........+ (5k + 4) + (5(k + 1) + 4 )  = [ (k + 1)/2] [5(k + 1) + 13 ]

9 + 14 + 19 + 80 +........+  (5(k + 1) + 4 )  = [ (k + 1)/2] [5(k + 1) + 13 ]

Which is what we wished to prove