For each integer n>1, let A(n) denote the sum of the integers from 1 to n. For example, A(100)=1+2+3+ +100=5,050. What is the value of A(200)?
A.)10,100
B.)15,050
C.)15,150
D.)20,100
E.)21,500
I know that the best answer choice is D, but the explanation that I was provided doesn't help in understanding the concept. If someone could provide an explanation and the topic area that this type of mathematics that this would fall under to get a better understanding of the concept, I would greatly appreciate it.
proof by induction
A little research will turn up many proofs that A(n) = n(n+1)/2
One easy way is to divide the numbers into pairs, from both ends.
1,n
2,n-1
3,n-2
...
Each pair adds up to n+1
There are n/2 such pairs.
The result follows.
Thanks, so this falls under induction by reasoning, if I'm not mistaken?
Thanks
To find the value of A(200), we need to calculate the sum of the integers from 1 to 200. This is essentially finding the sum of an arithmetic series.
The formula to find the sum of an arithmetic series is given by: S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term is 1 and the last term is 200. The number of terms, n, can be found by subtracting the first term from the last term and adding 1 (200 - 1 + 1 = 200).
Substituting the values into the formula, we have:
S = (200/2)(1 + 200)
= 100(1 + 200)
= 100(201)
= 20,100
Therefore, the value of A(200) is 20,100.
This type of mathematics falls under arithmetic and series. Specifically, it involves finding the sum of an arithmetic series.