  We survey the applications of an elementary identity used by Euler in one of
his proofs of the Pentagonal Number Theorem. Using a suitably reformulated
version of this identity that we call Euler's Telescoping Lemma, we give
alternate proofs of all the key summation theorems for terminating
Hypergeometric Series and Basic Hypergeometric Series, including the
terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz
sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of
Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$
sum. Our proofs are conceptually the same as those obtained by the WZ method,
but done without using a computer. We survey identities for Generalized
Hypergeometric Series given by Macdonald, and prove several identities for
$q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have
appeared in combinatorial contexts. Some of these identities appear to be new.
