A sequence of numbers, the method of mathematical induction

Definition: the Sequence —variable, depends on natural numbers (that is, a function of natural argument).

members (elements) of the sequence

If the items are real numbers, then the sequence is called a numeric

Examples

  • sequence of even natural numbers
  • is a sequence of integers negative numbers
  • — a sequence of numbers, reverse to natural
  • — numeric sequence

Ascending and descending sequences

Definition: a Sequence is called increasingif each subsequent term is greater than the previous: (the first sequence in the examples).

Definition: a Sequence is called decreasingif (second and third sequence in the examples).

The method of mathematical induction

Used to prove assertions about the number or sequence of expressions that depend on natural numbers in the language which explicitly or implicitly contains the words "for any natural "

Scheme of the proof of claims using the method of mathematical induction

  1. Check whether this statement is when (sometimes start with )
  2. Assume that the given statement holds for (the second option — when )
  3. Bring (based on assumption) the truth of our assertion and
  4. We conclude that this statement is true for any natural number (any )

Example.

To prove:

Rozwiazania. For recording convenience, we denote

  1. When the validity of the equation
  2. Assume that the given equality is correct if , that is,
  3. We prove that equality occurs when and , that is, prove that

    Given that , we obtain
  4. Therefore, the given equality is correct for any natural
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