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simcan18
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Mathematical induction is a method of mathematical proof used to prove that a statement is true for all values of a specific mathematical variable. It involves proving that the statement is true for the first value of the variable, and then showing that if the statement is true for any arbitrary value, it must also be true for the next value. This process is repeated until it can be shown that the statement is true for all values of the variable.
The formula being proven is Σ(1/[(2k-1)(2k+1)]=n/(2n+1), where Σ represents the summation symbol, k is the variable, and n is the number of terms in the series.
First, the formula is shown to be true for the first value of k (usually k = 1). Then, it is assumed to be true for an arbitrary value of k (called the "inductive hypothesis"). Using this assumption, the formula is then shown to be true for the next value of k. This process is repeated until it is shown that the formula is true for all values of k.
Mathematical induction is considered a valid method of proof because it is based on the fundamental mathematical principle that if a statement is true for a starting value and can be shown to be true for the next value based on the previous value, then it must be true for all values. This principle is known as the principle of mathematical induction.
Mathematical induction can be used to solve problems that involve proving a statement for all values of a specific variable. This can include problems related to sequences, series, and equations. To use mathematical induction, you will need to identify the starting value of the variable, the inductive hypothesis, and the method for showing that the statement is true for the next value of the variable. With practice, you can become proficient in using mathematical induction to solve various mathematical problems.