<p>In the xy-plane, the line 2x-3y = c passes through point (5, -1). What is the value of c?</p>
<p>Oh, also:</p>
<p>1,2,2,3,3,3,4,4,4,4...</p>
<p>All positive integers appear in the sequence above, and each positive integer k appears in the sequence k times. In the sequence, each term after the first is greater than or equal to each of the terms before it. If the integer 12 first appears in the sequence as the nth term, what is the value of n?</p>
<p>Answer: 67</p>
<p>Can some explain this to me? Thanks.</p>
<p>Thanks.</p>
<p>haaaaaaaaaaaaaaaalp</p>
<p>Yeesh, don't be so impatient.</p>
<p>"In the xy-plane, the line 2x-3y = c passes through point (5, -1). What is the value of c?"</p>
<p>2x-3y = c
(5, -1) is (x, y)
2(5) - 3(-1) = c
10 + 3 = c
13 = c</p>
<p>1 appears as the first term, appears once
2 appears as the second term, appears twice
3 appears as the fourth term, appears three times
4 appears as the 7th term, appears 4 times
5 appears as the 11th term, appears 5 times
6 appears as the 16th term, appears 6 times
7 appears as the 22nd term, appears 7 times
8 appears as the 29th term, appears 8 times
9 appears as the 37th term, appears 9 times
10 appears as the 46th term, appears 10 times
11 appears as the 56th term, appears 11 times
12 appears as the 67th term</p>
<p>Or, more simply
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 +11 +1<a href="since%20you%20start%20at%20the%20first%20term">/b</a> = **67</p>
<p>ya. just figure out the pattern and brute force it. It should take 20 sec tops on calc but it is sure to work.</p>
<p>the first one was too easy. no offense to anyone. :) just plug in the numbers.</p>
<p>In the xy-plane, the line 2x-3y = c passes through point (5, -1). What is the value of c?</p>
<p>In this problem, there are three variables. We know two of them (x=5, y=-1) so we can solve for the third.
2x-3y = 2(5) - 3(-1) = 10 +3 =13.</p>
<p>c = 13. </p>
<p>All positive integers appear in the sequence above, and each positive integer k appears in the sequence k times. In the sequence, each term after the first is greater than or equal to each of the terms before it. If the integer 12 first appears in the sequence as the nth term, what is the value of n?</p>
<p>There is a trick some savvy people know which makes this problem extremely easy. It is the sum formula for the function f(n) = n from n = 1 to n = k for some number k. It is given by S(k) = (1/2)(k)(k+1) </p>
<p>For example, S(4) = 1+2+3+4 = 10 = (1/2)(4)(5) = 10.</p>
<p>If we want to know how many terms are in the sequence from k=1 to k=11, we take the sum S(k) for k = 11, which is (1/2)(11)(12) = 66. </p>
<p>Add 1 to this because we want to know the term number of the first 12 in the sequence. </p>
<p>66 + 1 = 67.</p>