American settlement in the Oregon Country underwent significant changes starting in the 1840s, primarily due to westward expansion and the increased migration of settlers. Here are how the options you provided relate to these shifts:
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Fewer settlers came as the California Gold Rush drew most settlers: This statement is somewhat misleading. While the California Gold Rush in 1849 did attract a large number of settlers to California, it did not halt migration to the Oregon Country. In fact, many settlers moved to Oregon in the years leading up to the Gold Rush as they sought new opportunities and land. The Oregon Trail saw a surge in settlers during the 1840s, even with the Gold Rush occurring later in the decade.
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As the British established control over the region fewer Americans came: This is not accurate in the context of the 1840s. While there were ongoing disputes between the United States and Britain over the Oregon Territory, American migration to Oregon actually increased during this period. The Oregon Trail became a major route for settlers during the 1840s, and the idea of "Manifest Destiny" fueled the belief that Americans were destined to expand westward.
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Missionaries and settlers joined the fur traders who went there in early years: This statement is the most accurate. In the early years prior to the 1840s, fur traders were among the first Americans in the Oregon Country. By the 1840s, however, there was a significant influx of missionaries and settlers, as well as an increase in organized migration along the Oregon Trail. The establishment of missionary outposts and the promotion of the land's agricultural potential drew many settlers to the region.
Overall, the 1840s saw an increase in American settlement in the Oregon Country, fueled by a combination of factors including the desire for land, the availability of agricultural opportunities, and the influence of missionaries. Therefore, the most accurate representation of the changes in American settlement in the Oregon Country during this time would be that missionaries and settlers joined the fur traders who had previously entered the region.