Question

I'm in 7th grade and have to find an expression for the following: Door bell rings, 1 guest there, door bell rings again, 2 more than last time, door bell rings again, 2 more than last time, and so on. I have to determine how many guests would be there after 20th doorbell. I know the answer is 400, but need the algebraic expression. Can anyone help?

Answer 1

Someone please help!!

Answer 2

Look at the following:
term number term name term term as sum of a1
1 a1 1 a1 + 0 = a1 + 0
2 a2 3 a1 + 2 = a1 + 1(2)
3 a3 5 a1 + 4 = a1 + 2(2)
4 a4 7 a1 + 7 = a1 + 3(2)
...
n an n a1 + (n-1)2

With this, you can calculate the 20th term. a20 = a1 + (20 - 1)2 = 1 + (19)2 = 39
Now you know the first term is 1 and the last term is 39
Now think of this:
1, 3, 5, 7, 9, 11, 13, ... , 27, 29, 31, 33, 35, 37, 39
If you add the first and last terms, 1 + 39 = 40 Their average is 40/2 = 20
Now add second and next to last and get average: 3 + 37 = 40. Average = 20
Add 3rd therm and third from last: 5 + 35 = 40. Average: 20
If you keep doing this, all the averages will be 20.
Since you have 20 terms with an average value of 20, the total is 20(20) = 400