Chapter 1.2, Problem 2EE

The figure below shows that the fourth triangular number, $10$, added to the fifth triangular number, $15$, produces the fifth square number, $25$.

a. Use a drawing to show that the fifth triangular number added to the sixth triangular number is the sixth square number.

b. Verify that the 50^th triangular number added to the 51^st triangular number is the 51^st square number. Hint: Use a numerical approach; don’t use a drawing.

c. Use nth-term formulas to verify that the sum of the nth triangular number and the $\left(n+1\right)st$ triangular number is always the square number ${\left(n+1\right)}^2$.