Question

A sequence of numbers \(a_1, a_2, a_3\) etc. is generated using the following algorithm: \(a_{n+1}=\left(a_n\right)^2+2\). Does the number 35 appear in the sequence if: (1) 0 appears in the sequence (2) \(a_3>a_2\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) Answer D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

E

Step-by-step solution

Examine the Recurrence Relation

The given sequence follows the relation \(a_{n+1} = (a_n)^2 + 2\). This means that each term is obtained by squaring the previous term and adding 2.

Analyze Condition 1

Condition 1 states that 0 appears in the sequence. So if \(a_k = 0\) for some \(k\), then \(a_{k+1} = 0^2 + 2 = 2\). Thus, once 0 appears, the sequence will become \{..., 0, 2, 6, 38, ...\} and will keep generating numbers greater than or equal to 2 (never becoming 35).

Analyze Condition 2

Condition 2 states that \(a_3 > a_2\). Since \(a_{n+1} = (a_n)^2 + 2\) and squaring any real number and adding 2 will always result in a value greater than the original number, this condition is universally true. So this condition doesn’t provide specific information that helps determine if 35 appears in the sequence.

Evaluate Both Conditions Together

Analyzing both conditions together doesn’t provide any new information that influences the occurrence of 35 because Condition 1 alone already guarantees that 35 cannot appear in the sequence.

Conclusion

Given the analysis above, neither Condition 1 nor Condition 2 alone or together show that 35 appears in the sequence.

Key concepts

sequence algorithm

In mathematical terms, a sequence algorithm generates a sequence of numbers following specific rules. In this problem, the rule is given by the formula: \[a_{n+1} = (a_n)^2 + 2\]. This formula states that to get the next number in the sequence, you take the current number (\(a_n\)), square it, and then add 2. For example, if the first number \(a_1\) was 1, then the next few numbers in the sequence would be:

• \(a_2 = 1^2 + 2 = 3\)
• \(a_3 = 3^2 + 2 = 11\)
• \(a_4 = 11^2 + 2 = 123\)

Understanding how to apply this rule is key to solving any problem related to the sequence.

recurrence relation

Recurrence relations are equations that express each term of a sequence as a function of the preceding ones. For this sequence, the recurrence relation is: \[a_{n+1} = (a_n)^2 + 2\]. This relation defines how the sequence evolves over each step. Given the initial term, every subsequent term can be computed. This forms the backbone of our sequence. A critical aspect to note here is that as you keep applying the function, the terms grow very rapidly due to the squaring operation. Even small initial values can lead to very large numbers quickly.

quantitative reasoning

Quantitative reasoning involves the application of mathematical concepts to solve problems. In this exercise, we use our understanding of sequences and recurrence relations to determine if a specific number (35) can appear in the sequence under given conditions. For Condition 1, if 0 appears in the sequence, the next number would be:

• \(a_{k+1} = 0^2 + 2 = 2\)

Continuing further would yield 2, 6, 38, etc. We quickly see that the number 35 cannot be generated. For Condition 2, we note that squaring any number and adding 2 always yields a number greater than the original, making this universally true. Combining both conditions still won’t make 35 appear.

GMAT problem-solving

GMAT problem-solving questions test your ability to apply logic and mathematical understanding to find solutions. This exercise is a perfect example: you've been given a sequence algorithm and need to evaluate conditions to determine if a specific number can appear. Analyzing each condition involves breaking down the problem:

• For 0 to lead to 35, we show the sequence diverges
• For the inequality, we prove it holds universally but doesn't help in finding 35

Combining logical steps ensures a thorough and confident solution. It's crucial to practice such problems to enhance problem-solving skills for the GMAT.