Use the principle of mathematical induction to show that the mathematical statement is true for all natural numbers n.Sn: 1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8n = 
What will be an ideal response?
| S1: | 1 ? 8   |
8 = 8 ?
Sk: 1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8k =
Sk+1: 1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8(k + 1) =
We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely
8(k+1), to both sides."
1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8k + 8(k + 1) =
+ 8(k + 1)1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8(k + 1) =
+ 
1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8(k + 1) =
1 ? 8 + 2 ? 8 + 3 ? 8 + . . . + 8(k + 1) =
We have shown that if we assume that Sk is true, and we add (8(k+1) to both sides of Sk , then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n.
Mathematics
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Estimate the error in the quadratic approximation of the given function at the origin over the given region.
A. |E(x, y)| ? 0.4122 B. |E(x, y)| ? 0.2748 C. |E(x, y)| ? 0.2061 D. |E(x, y)| ? 0.8244
Mathematics
Construct a truth table for the given statement and then determine if the statement is a tautology.( ~ p ? q) ? (p ? q)
A. 
Is not a tautology.
B. 
Is not a tautology.
C. 
Is a tautology.
D. 
Is not a tautology.
Mathematics
Provide an appropriate response.Explain the purpose of each "-" sign in these examples. (a) 4 - 3 
(c) -(-4)
What will be an ideal response?
Mathematics
Subtract.19 - (-14)
A. -33 B. 5 C. -5 D. 33
Mathematics