Quiz: Real Sequences — 9 questions

1. Which condition characterizes an arithmetic sequence with common difference $r$?

A. Each term is obtained by multiplying the preceding term by $r$.

B. Each term is obtained by adding $r$ to the preceding term.

C. Each term is obtained by raising the preceding term to the power $r$.

D. Each term is obtained by dividing the preceding term by $r$.

Answer: (B)

Explanation

An arithmetic sequence has a constant difference, so consecutive terms satisfy $u_{n+1}-u_n=r$. Multiplication by a fixed factor instead characterizes a geometric sequence.

2. An arithmetic sequence has $u_3=11$ and common difference $r=4$. What is $u_8$?

A. 35

B. 44

C. 27

D. 31

Answer: (D)

Explanation

Using $u_n=u_m+(n-m)r$ gives $u_8=11+(8-3)\times4=31$. The value $27$ results from using four steps instead of five, while the other values apply an incorrect operation.

3. Which condition characterizes a geometric sequence with common ratio $q$?

A. Each term is obtained by adding $q$ to the preceding term.

B. Each term is obtained by subtracting $q$ from the preceding term.

C. Each term is obtained by multiplying the index by $q$.

D. Each term is obtained by multiplying the preceding term by $q$.

Answer: (D)

Explanation

A geometric sequence satisfies $u_{n+1}=q\,u_n$, so each term is formed by multiplication by the fixed ratio. Adding a constant describes an arithmetic sequence rather than a geometric one.

4. Let a geometric sequence satisfy $u_0=3$ and $q=2$. What is the limit of $u_n$ as $n$ tends to infinity?

A. It tends to $+\infty$.

B. It tends to $3$.

C. It tends to $0$.

D. It does not exist because the ratio is positive.

Answer: (A)

Explanation

Since $u_n=3\times2^n$ and $2>1$, the sequence grows without bound, so its limit is $+\infty$. A limit of $0$ occurs for powers with $-1<q<1$, not for this ratio.

5. What defines an arithmetic sequence?

A. A sequence where the difference between consecutive terms is constant.

B. A sequence where the sum of terms up to a certain point is constant.

C. A sequence where the terms follow a quadratic pattern.

D. A sequence where each term is multiplied by a constant factor.

Answer: (A)

Explanation

An arithmetic sequence is characterized by a constant reason, which is the difference between consecutive terms. The other options describe geometric sequences, constant sums, or quadratic patterns, which are not defining features of arithmetic sequences.

6. What is the defining characteristic of an arithmetic sequence?

A. A sequence where the sum of terms forms a geometric pattern.

B. A sequence where each term is the square of the previous term.

C. A sequence where each term is multiplied by a constant reason r to get the next term.

D. A sequence where the difference between consecutive terms is constant.

Answer: (D)

Explanation

An arithmetic sequence is characterized by a constant difference, r, such that $u_{n+1} - u_n = r$. The other options describe geometric sequences, quadratic patterns, or unrelated patterns.

7. What is the primary purpose of bounds and monotonicity in analyzing sequences?

A. To determine whether a sequence converges or diverges based on its upper and lower limits.

B. To establish the relationship between the sequence's terms and its sum.

C. To identify the initial term and common difference of an arithmetic sequence.

D. To calculate the exact value of each term in a sequence.

Answer: (A)

Explanation

Bounds and monotonicity help determine whether a sequence converges by analyzing its upper and lower limits, and whether it is increasing or decreasing. The other options focus on specific calculations or properties that are not directly related to the purpose of bounds and monotonicity.

8. When was the concept of bounds and monotonicity in sequences formally established in mathematical analysis?

A. In the 17th century, alongside the development of calculus.

B. In the early 20th century, with the formalization of set theory.

C. During the ancient Greek period, with the work of Euclid.

D. In the 19th century, as part of the development of real analysis.

Answer: (D)

Explanation

The formal study of bounds and monotonicity in sequences was developed in the 19th century as part of the rigorous foundation of real analysis. Earlier periods focused more on geometric and intuitive understandings, not the formal properties of sequences.

9. How do adjacent sequences help determine the limit of a sequence that is bounded between them?

A. Adjacent sequences always diverge unless they are constant.

B. If the outer sequences diverge, the sequence in between must also diverge.

C. Adjacent sequences only provide bounds if they are monotonic.

D. If the outer sequences converge to the same limit, the sequence in between also converges to that limit.

Answer: (D)

Explanation

When two sequences are adjacent, with one increasing and the other decreasing, and both converge to the same limit, they effectively trap the sequence in between, forcing it to also converge to that limit. If the outer sequences diverge, the sequence in between may also diverge, so the key is their common limit.