How do you find the sum of the arithmetic series #6+13+20+27+...+97#?

1 Answer
Oct 19, 2016

#S_14 = 721#

Explanation:

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#