To solve this problem, we need to identify the sentence that does not fit cohesively with the other sentences to form a coherent paragraph related to music's connection to human identity and cognition. Let's analyze the given sentences:
Now, let's identify the thematic flow: sentences 1, 3, 4, and 5 cohesively discuss the universality of music, its connection to human identity, its demanding nature cognitively, and its presence across various cultures. Thus, they form a coherent paragraph.
Sentence 2: "The profound emotional impact of music has inspired ongoing research into its relationship with emotions." This sentence brings in the specific aspect of emotional impact and associated research, which diverts from the discussion of music's universality and cognitive aspects. It introduces a new element that doesn't align with the central theme formed by the other sentences.
Conclusion: Hence, the sentence that does not fit with the rest of the paragraph is Sentence 2. The correct answer is:
The profound emotional impact of music has inspired ongoing research into its relationship with emotions.
To identify the odd sentence out, we need to arrange the sentences in a logical sequence that creates a coherent paragraph, and then determine which sentence does not fit this theme or flow. Let us examine each sentence:
From these sentences, we can see that the common theme among sentences 2, 3, 4, and 5 is the pervasive and complex nature of music in human life. They collectively describe music's universality, its cognitive demand, and its role in human identity and society.
Now, let’s evaluate the sentences:
Thus, sentence 1 does not fit seamlessly with the others as it pivots to a different aspect—research concerning emotions, separated from the more direct discussion of music’s role and impact on human culture and cognition.
Conclusion: The odd sentence out is sentence 1: "The profound emotional impact of music has inspired ongoing research into its relationship with emotions."
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: