snaidu228 said:Homework Statement
If x is a real number, we de fine [x] as being the largest integer <= x. For example, [1.2] = 1,
[-1.1] = -2, [1] = 1, [11/3]3 = 3, . . .
Let {an}n>=1 be the numerical sequence de fined by:
a1 = 3; and an = a[n/2], for n>=2
(a) Give the terms a1; a2; ... ; a8 of this sequence.
(b) Prove that an = 3; For all n>= 1
Homework Equations
The Attempt at a Solution
I'm not sure what induction has to do with this... I don't really get it.
The purpose of finding the terms and proving induction for a numerical sequence is to determine the pattern or rule that governs the sequence and to prove that the rule holds true for all numbers in the sequence. This allows for accurate prediction and calculation of future terms in the sequence.
The first step is to observe the given numerical sequence and identify any patterns or rules that may be present. Then, based on these observations, a general formula for the nth term in the sequence can be formulated. Next, the formula can be tested by plugging in different values for n and checking if the resulting terms match the given sequence. Finally, mathematical induction can be used to prove that the formula holds true for all values of n.
Mathematical induction is a method of proof that involves showing that a statement is true for a base case (usually n=1) and then assuming that the statement is true for some arbitrary value of n (usually k) and proving that it is also true for the next value (n=k+1). If the statement holds true for both the base case and the next value, it can be concluded that it holds true for all values of n.
Proving the formula for a numerical sequence ensures that the rule or pattern identified is accurate and holds true for all numbers in the sequence. This allows for confidence in using the formula to predict and calculate future terms. It also provides a deeper understanding of the sequence and its properties.
Yes, mathematical induction can only be used to prove a formula for discrete numerical sequences, where each term is a whole number. It cannot be used for continuous sequences or sequences with non-numeric terms. Additionally, the formula must hold true for all values of n, so it may not be applicable to sequences with exceptions or outliers.