Preliminary Examination in Number Theory

What is the first term of the sequence defined by a_n= 2n + 1?

• 1

• 2

• 3

• 4

What is the sum of the first five positive integers?

• 10

• 15

• 20

• 25

What is the first step in a mathematical induction proof?

• Prove the case for n=k.

• Prove the case for n=k+1.

• Prove the base case n=1.

• Assume the statement is true for n=k.

What is the third term of the Fibonacci sequence, where F1=1 and F2=1?

• 1

• 2

• 3

• 5

What is the value of "4 choose 2"?

• 4

• 6

• 8

• 10

If a sequence is defined as a_n=n²-1, which of the following explains the relationship between consecutive terms?

• Each term is the square of the position minus one.

• The difference between consecutive terms increases by 2.

• Each term is obtained by subtracting 1 from the previous term.

• The terms alternate between positive and negative values.

Why is the formula shown valid for the sum of the first n positive integers?

• It is a simplification based on grouping pairs of terms with the same sum.

• It is derived by adding all the numbers and dividing by 2.

• It assumes that all numbers are multiplied by 2.

• It works only when n is even.

Why is it necessary to prove the base case in mathematical induction?

• To check that the formula works for all even numbers.

• To verify that the statement is true for the smallest possible value of n.

• To show that the assumption step is valid.

• To simplify the induction hypothesis.

Why is each term in the Fibonacci sequence the sum of the two previous terms?

• It represents a natural process of growth in population studies.

• It alternates between odd and even terms.

• It ensures that the sequence increases by a fixed amount.

• It follows the formula Fn=n(n-1).

What does nr represent in combinatorics?

• The sum of all integers from 1 to r.

• The number of ways to choose r objects from nn without regard to order.

• The product of all numbers from n to r.

• The total number of arrangements of n objects.