Answer :
The entries of the next row in Pascal's Triangle are 7, 21, 35, 35, 21, 7, 1.
To complete the arithmetic sequence, we need to find the missing terms using the common difference between each consecutive term. Let d be the common difference.
The first missing term is 3 less than 9, so it is 6. That is, the sequence is now: __, 9, __, __, 39, __
From 9 to the second missing term, there are four intervals of d, so the second missing term is 9 + 4d.
From the second missing term to the third missing term, there are two intervals of d, so the third missing term is (9 + 4d) + 2d, or 9 + 6d.
Finally, from the third missing term to 39, there are nine intervals of d, so we have:
(9 + 6d) + 9d = 39
Simplifying gives:
15d = 30
d = 2
So the completed sequence is: 5, 9, 11, 13, 39, 41
To complete the geometric sequence, we need to find the missing terms using the common ratio between each consecutive term. Let r be the common ratio.
Since -2 times the common ratio twice gives us 54, we have:
-2r^2 = 54
Dividing both sides by -2 gives:
r^2 = -27
This equation has no real solutions, since the square of any real number is nonnegative. Therefore, there is no way to complete this geometric sequence using real numbers.
To determine the entries of the next row in Pascal's Triangle, we can use the fact that each entry in the row is the sum of the two entries above it in the previous row, with the convention that the first and last entries in each row are 1.
Using this rule, we can determine the next row as follows:
1 6 15 20 15 6 1
7 21 35 35 21 7
Therefore, the entries of the next row in Pascal's Triangle are 7, 21, 35, 35, 21, 7, 1.
Learn more about Pascal's Triangle here:
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