This paper discusses Gosper and Zeilberger algorithms which are the cornerstone of proving the hypergeometric identities. Solving the hypergeometric identities that involve binomial coefficients, factorials, rational functions, and power functions is simplified immensely with computer algebra. Computer algebra started to develop with the Gosper algorithm for indefinite sums in 1978 and the Zeilberger algorithm for definite sums in 1990. I first analyze how the algorithms work explicitly. I then present a derivation of Gosper's algorithm which allows the generalization to high-order recurrence. Next I analyze how the computer finds whether or not a given hypergeometric sum is expressible in simple closed form. After some preliminary examples, I give the computerized short proofs of hypergeometric identities with the Maple.
