(a) To compute the values of a1, a2, a3, a4, and a5, we use the formula derived for by: by = an+1 - an = n - 2 Starting with a = 0, we have: a1 = a + b1 = 0 + (1 - 2) = -1 a2 = a + b2 = 0 + (2 - 2) = 0 a3 = a + b3 = 0 + (3 - 2) = 1 a4 = a + b4 = 0 + (4 - 2) = 2 a5 = a + b5 = 0 + (5 - 2) = 3
Therefore, the values of {an} are: 0, -1, 0, 1, 2, 3.
(b) The points with coordinates (n, an) for n = 0, 1, 2, 3, 4, and 5 are: (0, 0), (1, -1), (2, 0), (3, 1), (4, 2), and (5, 3). Plotting these points in the Cartesian plane gives the following graph:
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4 | o
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3 | o
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2 | o
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1 | o
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0 | o
|_________________________
0 1 2 3 4 5
(c) To find a formula for an, we can use the formula derived for by: by = an+1 - an = n - 2 Rearranging this equation, we get: an+1 = an + (n - 2) Substituting a = 0 and expanding the recurrence relation, we get: a1 = -1 a2 = a1 + (2 - 2) = -1 a3 = a2 + (3 - 2) = 0 a4 = a3 + (4 - 2) = 1 a5 = a4 + (5 - 2) = 3
Therefore, we can see that an = n(n-1)/2 for n >= 2, and a1 = -1.
(d) To check if the formula derived in part (c) gives the correct values of a, we plug in n = 1, 2, 3, 4, and 5: a1 = -1, a2 = 0, a3 = 1, a4 = 2, a5 = 3. These values match the values found in part (a), so we can be confident that the formula is correct for all values of n.
(a) To compute the values of a1, a2, a3, a4, and a5, we use the formula derived for by: by = an+1 - an = n - 2 Starting with a = 0, we have: a1 = a + b1 = 0 + (1 - 2) = -1 a2 = a + b2 = 0 + (2 - 2) = 0 a3 = a + b3 = 0 + (3 - 2) = 1 a4 = a + b4 = 0 + (4 - 2) = 2 a5 = a + b5 = 0 + (5 - 2) = 3
Therefore, the values of {an} are: 0, -1, 0, 1, 2, 3.
(b) The points with coordinates (n, an) for n = 0, 1, 2, 3, 4, and 5 are: (0, 0), (1, -1), (2, 0), (3, 1), (4, 2), and (5, 3). Plotting these points in the Cartesian plane gives the following graph:
|
4 | o
|
3 | o
|
2 | o
|
1 | o
|
0 | o
|_________________________
0 1 2 3 4 5
(c) To find a formula for an, we can use the formula derived for by: by = an+1 - an = n - 2 Rearranging this equation, we get: an+1 = an + (n - 2) Substituting a = 0 and expanding the recurrence relation, we get: a1 = -1 a2 = a1 + (2 - 2) = -1 a3 = a2 + (3 - 2) = 0 a4 = a3 + (4 - 2) = 1 a5 = a4 + (5 - 2) = 3
Therefore, we can see that an = n(n-1)/2 for n >= 2, and a1 = -1.
(d) To check if the formula derived in part (c) gives the correct values of a, we plug in n = 1, 2, 3, 4, and 5: a1 = -1, a2 = 0, a3 = 1, a4 = 2, a5 = 3. These values match the values found in part (a), so we can be confident that the formula is correct for all values of n.
