1992 AIME Problem 8

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

A sequence with constant second difference

1

is given by a quadratic with leading coefficient

\frac{1}{2}

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

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.

Problem 8 in other years