Some Fascinating Sum Formula Involving Balancing Number

Abstract

The problem of finding sums of specific sequences of natural numbers had been a fascination to mathematicians. In this connection, there is an interesting story about the famous German mathematician Carl Friedrich Gauss when he was just eight years old and was in primary school. One day Gauss' teacher asked his class to add together all the numbers from 1 to 100, assuming that this task would occupy them for quite a while. He was shocked when young Gauss, after a few seconds thought, wrote down the answer 5050. There are binary recurrence sequences having sum formulas resembling that for natural numbers, e.g. the sum of first n odd positive integers is equal to n^2, while the sum of first n odd balancing numbers is equal to the square of the nth balancing numbers. The talk focuses on certain sum formulas involving balancing numbers. In each formula, if the balancing numbers is replaced by its index, it reduces to a known formula for natural numbers.