Triangular Numbers

Question

T(n)=the sum of the numbers of ai as i runs from 1 to n.

a) write the sum in longhand form without using summation notation.

b) perform the addition to calculate T(n) for n=1,2,3,4,5.

c) find the values of T(2) + T(1), T(3)+ T(2), and T(4) + T(3). What property do these three numbers have in common?

e) generalize the result in part (d), by finding the value of T(n) + T(n-1).

f) find the value of T(n) - T(n-1).

g) create a formula for T(n) from e and g.

h) write the formula for T(n-1).

Answer 1

This is very nice exercise in finite differences and it should not require much help from a tutor like me if you just follow along the logic of the questions.

Note, however, that this is NOT the usual definition of triangular numbers.

The critical answer is part g, for which you should get T(n)=n(n+1)/2...and historically you should read the story about C. F.Gauss, one of the greatest of all time mathematicians.

To get part g:

write S=1+2+3...+n

then write the sum backwards, S=n+(n-1)+....+1

Then add the 2 expression for 2.

If you are really stuck, make a comment which I will receive and I will try to be of further help.

S=1+2+3+..... +(n-1)+n

S=n+(n-1) +...+1

2S= (n+1)+(n+1).....+(n+1) and there are n terms

Therefore, S=T(n)=(1/2)n(n+1)

OK, let's agree that T(n)=n(n+1)/2...at least I hope you got that; if lot comment again.

T(n+1)+T(n)=(1/2)[n²+n+n²+3n+1]=n²+2n+1...so the some of any 2 successive sums is a perfect square

The difference between T(n) and T(n-1) = n....just do the subtraction.