Question

For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots),$ define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots),$ whose $n$ th term is $a_{n+1}-a_n.$ Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1,$ and that $a_{19}=a_{92}=0.$ Find $a_1.$

Answer

Answer: 819

Concepts:quadratic recursion factoring

Difficulty rating: 1980

Small Hint:

A sequence with constant second difference $1$ is given by a quadratic with leading coefficient $\frac{1}{2}$

Big Hint:

Use the two zero terms to write the quadratic in factored form

Solution:

A quadratic sequence with second difference $1$ has leading coefficient $\frac{1}{2}.$ Since its values vanish at indices $19$ and $92,$ $a_n=\frac12(n-19)(n-92).$ Therefore $a_1=\frac12(-18)(-91)=819.$

1992 AIME Problem 8

Answer: 819

Difficulty rating: 1980

Small Hint:

Big Hint:

Solution:

Problem 8 in Other Years