# 2008 iTest Problems/Problem 42

## Problem

Joshua's physics teacher, Dr. Lisi, lives next door to the Kubiks and is a long time friend of the family. An unusual fellow, Dr. Lisi spends as much time surfing and raising chickens as he does trying to map out a $\textit{Theory of Everything}$. Dr. Lisi often poses problems to the Kubik children to challenge them to think a little deeper about math and science. One day while discussing sequences with Joshua, Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776,-1765,-1754,\ldots$. Joshua then computes the (positive) difference between the $1980^\text{th}$ term in the sequence, and the $1977^\text{th}$ term in the sequence. What number does Joshua compute?

## Solutions

### Solution 1

To get from the $1977^\text{th}$ term to the $1978^\text{th}$ term, add $11$. To get from the $1978^\text{th}$ term to the $1979^\text{th}$ term, add $11$. To get from the $1979^\text{th}$ term to the $1980^\text{th}$ term, add $11$. Thus, the (positive) difference between the $1980^\text{th}$ term in the sequence and the $1977^\text{th}$ term in the sequence is $\boxed{33}$.

### Solution 2

The explicit formula of the arithmetic series is $$a_n = -1776 + 11(n-1)$$ Using this formula, the $1977^\text{th}$ is $-1776 + 11 \cdot 1976$ and the $1980^\text{th}$ term is $-1776 + 11 \cdot 1979$. The difference between the two terms is $-1776 + 11 \cdot 1979 + 1776 - 11 \cdot 1976 = \boxed{33}$.