We have #" "color(red)(6)+13+20+27+......+color(blue)(97)#
We have a formula for the sum of an arithmetic series:
#S_n = (n(color(red)(a_1)+color(blue)(l)))/2#
We have the first term and the last term, but not the number of terms. We need to find that first:
For this series, we know:
#color(red)(a_1 = 6), " " color(darkviolet)(d= 7) " " T_n = a_1 + (n-1)d " and " color(blue)(T_n= 97)#
#T_n = color(red)(6) + (n-1)color(darkviolet)(7) = color(blue)(97)" "larr# solve for n
#6 +7n -7 = 97#
#7n = 97+1#
#7n =98#
#color(lime)(n = 14)" "larr# now we can find the sum of 14 terms
#S_color(lime)(n) = (color(lime)(n)(color(red)(a_1)+color(blue)(l)))/2#
#S_color(lime)(14) = (color(lime)(14)(color(red)(6)+color(blue)(97)))/2#
#S_14 = 7(103)#
#S_14 = 721#
