A student earns $100 in the first week of June. Each week the student earns $12.50 more than the previous week. After twelve weeks, how much has the student earned?

Question

(Use sigma notation for partial sums)

Answer 1

total earnings for 12 weeks = $2,025

it's an arithmetic sequence with common difference = d = $12.50

and initial term = $100

12th term = 100+12.5(12-11) = $237.50

sum of 12 terms = (12/2)(100+237.5) = $2025

credit goes to Carl Friedrich Gauss for the formula

he was in maybe the 3rd grade when he came up with it

Sn = (n/2)(a1+a12) for the partial sum of the 1st through nth terms

he got an A in his arithmetic class

use a capital Greek letter Sigma, which corresponds to our English S

standing for Sum with an i=1 at the bottom and a 12 at the top of the Sigma

then to the right is ai

100 1st week

112.50 partial sum = 212.5

125 3rd week partial sum = 337.5

137.50 4th week partial sum = 475

150 5th week partial sum = 625

162.5 6th week partial = 787.5

175 7th week, partial sum = 962.5

187.5 8th week, partial = 1150

200 9th week, partial = 1350

212.5 10th week, partial 1562.5

225 11th week, partial 1787.5

237.50 12th week partial $2,025

Sn = n/2(a1+an)

for n=12

S12 = 12/2(a1+a12)

= 12/2(100+237.5)

=6(337.5)

=$2,025

an = a1 + d(n-1) where d= the common difference = 12.5

for n=12, the 12th term is:

a12 = 100 + 12.5(12-1)

a12 = 100+12.5(11)

a12 = 100 + 137.5

a12 = 237.5