# Thread: sum of arithmetic sequence

1. ## sum of arithmetic sequence

A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce is increased by 3 people.
How many people were there in the workforce after 26 weeks (I answered 89, which is correct.)
Each member of the workforce earned $600 per week. What was the total wage bill for this 26 weeks? I used the formula for the sum of an arithmetic sequence S=0.5n(a+l) and (for convention) 'eliminated' every other week where the wage bill remained the same to come up with S(half the wage bill) = 0.5 x 13 x (30 000 + 51 600) So the total wage bill is 13 x 81 600 =$ 1 060 800

However the textbook gives the answer to be $1 341 600 Where did I go wrong? 2. ## Re: sum of arithmetic sequence Hello Furor, I agree with your result for the wages. I think the result given in the text book is incorrect. Here is how it is I would like to suggest the following observation wages Weeks 30000 1-2 30000 +1800 2-4 30000 +2*1800 4-6 : : : sum of wages = 26*(30000) + 3600*(0+1+....+12) = 780000+ 3600*78 = 1060800 Now one thing that we can fairly agree is the fact that the 26*30000 component is the component that has to be paid as 50 workers are present from the beginning Lets deduct that from both the numbers 1060800 - 780000 = 280800 1341600 - 780000 = 561600 So we see that 561600 = 2*280800 which essentially tells us that a miscalculation has been done in terms of halving the number of weeks. Try to verify my calculations in case I made a mistake some where. Kalyan 3. ## Re: sum of arithmetic sequence Originally Posted by furor celtica A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce is increased by 3 people. How many people were there in the workforce after 26 weeks (I answered 89, which is correct.) Each member of the workforce earned$600 per week. What was the total wage bill for this 26 weeks?

I used the formula for the sum of an arithmetic sequence S=0.5n(a+l) and (for convention) 'eliminated' every other week where the wage bill remained the same to come up with S(half the wage bill) = 0.5 x 13 x (30 000 + 51 600)
So the total wage bill is 13 x 81 600 = $1 060 800 However the textbook gives the answer to be$ 1 341 600
Where did I go wrong?
Hello!
For the first part of the question, I assume you did 50 + (3 people x 13 weeks) = 89 people, because every 2 weeks there is an increase of 3 people. Which is correct.

Next part, because I have no idea the arithmetic sequence, i'll use the long method.

50 people x 600$x 26 weeks = 780,000 I did this because 50 people is a constant. 3 people x 600$ x 24 weeks = 43200
3 people x 600$x 22 weeks = 39600 3 people x 600$ x 20 weeks = 36000, Now the pattern decreases by 3600 so..
Week 18 = 32400
Week 16 = 28800
Week 14 = 25200
Week 12 = 21600
Week 10 = 18000
Week 8 = 14400
Week 6 = 10800
Week 4 = 7200
Week 2 = 3600

Add the numbers to get .. 280800
Add the 2 numbers to arrive at 280,800 + 780,000 = \$1 060 800.

Either the book is wrong, or you and I both did something wrong. Good luck.