91Ó°ÊÓ

Let \(t_{n}\) denote the \(n\) th triangular number. Find an explicit formula for \(t_{n}.\)

Let \(A=\left(a_{j}\right)\) and \(B=\left(b_{i j}\right)\) be two \(n \times n\) matrices. Let \(f_{n}\) denote the number of computations (additions and multiplications) to compute their product \(C=\left(c_{i j}\right),\) where \(c_{i j}=\sum_{k=1}^{n} a_{k} b_{k j}\) Evaluate \(f_{n}\)

Prove each, where \(F_{n}\) is the \(n\) th Fibonacci number, \(L_{n}\) the \(n\) th Lucas number, and \(\alpha=(1+\sqrt{5}) / 2,\) the golden ratio. $$F_{n}^{2}-F_{n-1} F_{n+1}=(-1)^{n-1}, n \geq 2$$

Let \(t_{n}\) denote the \(n\) th triangular number. Prove that \(8 t_{n}+1\) is a perfect square.

Let \(A=\left(a_{i j}\right)\) and \(B=\left(b_{i j}\right)\) be two \(n \times n\) matrices. Let \(f_{n}\) denote the number of computations (additions and multiplications) to compute their product $$C=\left(c_{i j}\right), \text { where } c_{i j}=\sum_{k=1}^{n} a_{i k} b_{k j}$$ Evaluate \(f_{n}\)

Let \(t_{n}\) denote the \(n\) th triangular number. Prove that \(8 t_{n}+1\) is a perfect square.

Let \(A=\left(a_{j}\right)\) and \(B=\left(b_{i j}\right)\) be two \(n \times n\) matrices. Let \(f_{n}\) denote the number of computations (additions and multiplications) to compute their product \(C=\left(c_{i j}\right),\) where \(c_{i j}=\sum_{k=1}^{n} a_{k} b_{k j}\) Estimate \(f_{n}\)

Let \(r_{n}\) and \(s_{n}\) be two solutions of the recurrence relation \((5.8) .\) Prove that \(a_{n}=r_{n}+s_{n}\) is also a solution.

Let \(A=\left(a_{i j}\right)\) and \(B=\left(b_{i j}\right)\) be two \(n \times n\) matrices. Let \(f_{n}\) denote the number of computations (additions and multiplications) to compute their product $$C=\left(c_{i j}\right), \text { where } c_{i j}=\sum_{k=1}^{n} a_{i k} b_{k j}$$ Estimate \(f_{n}\)

Prove each, where \(F_{n}\) is the \(n\) th Fibonacci number, \(L_{n}\) the \(n\) th Lucas number, and \(\alpha=(1+\sqrt{5}) / 2,\) the golden ratio. \(F_{5 n}\) is divisible by \(5, n \geq 1\)