Mitchell plans to work at a day camp over the summer. Each week, he wi

Question

Mitchell plans to work at a day camp over the summer. Each week, he will be paid according to the following schedule: at the end of the first week, he will receive $1. At the end of each subsequent week, he will receive $1, plus an additional amount equal to the sum of all payments he’s received in previous weeks. How much money will Mitchell be paid in total during the summer, if he works for the entire duration of the 8-week-long camp?

A. 126
B. 127
C. 252
D. 254
E. 255

A. 126
B. 127
C. 252
D. 254
E. 255

Archit3110 · Answer

WeekCurrent Week Pay Previous Week pay Total pay of the Week
Week 1 $1 $0 $1
Week 2 $1 $1 $2
Week 3 $1 $3 $4
Week 4 $1 $7 $8
Week 5 $1 $15 $16
Week 6 $1 $31 $32
Week 7 $1 $63 $64
Week 8 $1 $127$128

total $255 IMO E

baru · Answer

Paid amount for Week 1 = 1 ,
 Week 2 = 1+1 = 2,
 week 3 = 1+2+1 = 4 ( 2*2 , week2 *2)
 week 4 = 1+2+4+ 1 = 8 ( 4*2 , Week3 *2)
similary pattern is 1,2,4,8,16,32,64,128

So week 8 = 128 , instead of adding all here for the Total amount , 
  we can do ( week 9 -1 ) = 128*2 -1 = 255 ( As give namount paid per week = Total paid till then + 1 )

ScottTargetTestPrep · Answer

Solution:

We can tabulate the amount Mitchell earns each week.

Week 1: $1

Week 1: 1 + 1 = $2

Week 3: 1 + (1 + 2) = $4

Week 4: 1 + (1 + 2 + 4) = $8

At this point, we see that the amount he earns in certain week is double the amount he earns the previous week. Therefore,

Week 5: $16 Week 6: $32 Week 7: $64 Week 8: $128

So the total he earns at the end of the 8-week-long camp is:

1 + 2 + 4 + … + 128 = 2^8 - 1 = $255

(Note: We are using the formula 1 + 2 + 4 + … 2^n = 2^(n + 1) - 1 and notice that 128 = 2^7.)

Answer: E

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Mitchell plans to work at a day camp over the summer. Each week, he wi

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