SOLUTION: On the first day, Joe joined a secret club as its only member. Each day after that, one more member joined than on the previous day. What was the membership of the club after 40 da

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Question 258380: On the first day, Joe joined a secret club as its only member. Each day after that, one more member joined than on the previous day. What was the membership of the club after 40 days?
((( I did this: 40 multiplied by 2 ; because it says each day after one more member joined than on the previous. So then I got 80 and added joe and got 81???)))
Found 2 solutions by richwmiller, Earlsdon:
Answer by richwmiller(17219) (Show Source):
You can put this solution on YOUR website!
day joined membership
0 0 0
1 1 1
2 2 3
3 3 6
4 4 10
5 5 15
6 6 21
7 7 28
8 8 36
9 9 45
10 10 55
11 11 66
12 12 78
13 13 91
The new total is the previous total plus the new day
#8 is 36 which is 1/2*8*9
#10 is 55 which is 1/2*10*11=55
#12 78 1/2 12*13=6*13
total = day*day+1*1/2
It also works with odd numbers
#11 11*12*1/2=66
so we have
t=d*(d+1)*1/2
t=40*41*1/2
t=20*41=820 after 40 days

Answer by Earlsdon(6294) (Show Source):
You can put this solution on YOUR website!
Well, it doesn't quite work that way. You can easily see this when realize that on the 40th day, 40 new members were added and on the previous day, 39 new members were added, so you already have 79 new members added in just the last two days.
The series of numbers is called an arithmetic series.
You could take the approach that mathematician Gauss is said to have taken when, as a lad of 10 years, his math teacher asked the class to add up all of the integers from 1 to 100, hoping to keep them busy for a while.
It is said that Gauss arrived a the correct solution in a matter of seconds(?).
He added the first and last integer (1+100) to get 101.
Then he added the second and next-to-last number (2+99) to get 101.
He did this for the 50 pairs of integers and he got 50 X 101 = 5,050.
Try this on your membership problem.
1+40 = 41
2+39 = 41 etc for 20 pairs.
Now 20 X 41 = 820 members.
The formula for this is called the partial sum formula:
where: is the partial sum of n terms, is the first term and is the nth term.