the natural numbers are grouped as follows:
(1) , ( 2 , 3 ) , ( 4 , 5 , 6 ) , ( 7 , 8 , 9 , 10 ) , ...
find an expression for the first term of the nth group and prove that the sum of the numbers in the nth group is 1/2 n (n^2+1).
uh...I have no idea where to even start.
thank you in advance!
Hello, colloquial!
The natural numbers are grouped as follows:
(a) Find an expression for the first term of the group.
(b) Prove that the sum of the numbers in the group is: .
Note the last number of each group: .
. . These are the triangular numbers: .
So each group begins with the preceding triangular number, plus 1.
(a) The first term of the group is: .
(b) Each group is an arithmetic series with first term, ,
. . common difference, , and terms.
The sum of an arithmetic series is: .
Therefore: .