Question - Finding the Next Terms in Number Sequences
Solution:
The image shows three sequences where the question is asking to find the next three terms of each.For the first sequence (19, 23, 29,...), it looks like the sequence of prime numbers starting from 19. The next three prime numbers after 29 are: 31, 37, and 41.For the second sequence (15, 24, 36,...), we can find the differences between consecutive terms to try and detect a pattern: 24 - 15 = 9, and 36 - 24 = 12. It seems like there might be a pattern of increasing by 3 more than the previous increase each time. If this pattern continues, the next difference would be 12 + 3 = 15. Adding this to the last term (36 + 15), we get the next term of 51. Continuing the pattern further, we get the next difference as 15 + 3 = 18, and adding this to 51 gives us 69. Once again, 18 + 3 = 21, added to 69 gives us 90. So the next three terms should be 51, 69, 90.For the third sequence ($$ \frac{1}{2}, \frac{2}{9}, \frac{3}{17},\ldots $$), the numerators are increasing by 1 each time (1, 2, 3,...), and the denominators seem to be increasing by an increment, which increases by 1 each time as well (2, 9, 17,...). If we examine the denominators: 9 - 2 = 7, and 17 - 9 = 8. This suggests we're adding one more than the previous time to the denominators. So the next difference for the denominator would be 8 + 1 = 9; therefore, 17 + 9 = 26. Thus, the next term would be $$ \frac{4}{26} $$, but this can be simplified to $$ \frac{2}{13} $$ because both the numerator and denominator can be divided by 2.Continuing this pattern, for the following term, we need to add 9 + 1 = 10 to the previous denominator of 26, giving us 26 + 10 = 36. The fifth term should be $$ \frac{5}{36} $$.For the next term, we add 10 + 1 = 11 to the previous denominator, so 36 + 11 = 47. The sixth term should be $$ \frac{6}{47} $$. Therefore, the next three terms are $$ \frac{2}{13}, \frac{5}{36}, \frac{6}{47} $$.In summary, the next three terms for each sequence are:1. 31, 37, 412. 51, 69, 903. $$ \frac{2}{13}, \frac{5}{36}, \frac{6}{47} $$
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