<p>can anyone show me how to solve this x_x</p>
<p>what is the sum of three-digit positive multiples of 17?</p>
<p>thanks</p>
<p>all the 3-digit multiples? Sounds USAMO to me… but in any case, all you do is find the first number x where 17 times x greater than 100, which is 6 in this case, and then the last number y where 17 times y is less than 1000. This number is 58. If you have a ti-89 you can just do a summation, with 17x as the formula to plug into.</p>
<p>You might be able to do that on a ti-83/84, but I wouldn’t remember. :P</p>
<p>i don’t think it’s that hard
but anyway you’d take the first multiple of 17 above 99 (102) add it to the last multiple above 99 (986), add them together (1088), multiply by the number of multiples between 100 and 999 ((986-102)/17+1) and divide by two and i got 28832, but i might be wrong</p>
<p>I agree with karkaputto:</p>
<p>We want the sum of 6 x 17 through 58 x 17,
which is equal to 17 times the sum of 6 through 58;
the sum of 6 through 58 is equal to (58-6+1) times the average of 6 and 58;</p>
<p>= 17 x 1696</p>
<p>= 28,832</p>
<p>(about Mathcounts chapter team level)</p>
<p>i’ve always wondered how to do the sum of multiple consecutive numbers.</p>
<p>So the formula is what? The highest number - the lowest number + 1, times the average of the lowest and highest number?</p>
<p>thx!
hm… sorry but
another question…</p>
<p>how would u solve this:
what is the sum of the multiples of 13 from -500 to 1500, inclusive?</p>
<p>Let me try…the sum of all multiples of 13 between -500 to -1 and 1 to 500 cancel out. But since its inclusive, you still have that -500 there.</p>
<p>the sum of all multiples of 13 between 500 and 1500 is 989,989. But you still have that 1500 there since its inclusive.</p>
<p>So its… 989,989-500+1500= 990,989.</p>
<p>someone correct me if i’m wrong.</p>
<p>S2530S2:
CAA5042 is right in that the sum of multiples of 13 from (-500 to -1 )and (1 to 500) cancel out. The question is effectively ‘What is the sum of the multiples of 13 from 501 to 1500, inclusive?’</p>
<p>501/13 = 38.54 and 1500/13 = 115.38 . So, the answer is</p>
<p>(13)( 39 + 40 + 41 + … + 114 + 115)
= (13) (115-39 + 1) (39+115) / 2
= (13) (77) (77)
= 77077</p>
<p>(CAA5042: In your post#8, you forgot to include the logic for ‘multiples of 13’ , I think.)</p>
<p>i see u included 40 in (13)( 39 + 40 + 41 + … + 114 + 115)</p>
<p>but 40 isn’t a multiple of 13…</p>
<p>yeah optimizer- just noticed that.But what happened to -500? I thought that wasn’t supposed to cancel out with 500 since 500 isn’t a multiple of 13? In other words,aren’t we supposed to omit 500(since its not a multiple of 13) and leave -500 there(since its inclusive)? I’m not sure though.</p>
<p>S2530S2- i think the numbers in the summation represent numbers which when multiplied by 13, gives you a number between 500 and 1500… If that makes sense. I cant really articulate it so lets wait for optimizer.</p>
<p>that’s not what “inclusive” means…
inclusive means that if 1500 was a multiple of 17, it would be counted as opposed to non-inclusive which would mean that if 1500 was a multiple of 17, it still would not be counted
i guess the math folks aren’t too good at english</p>
<p>lol- was that comment really necessary? We all learn something new each day.</p>
<p>The sum of multiples of 13 from (-500 to -1 )and (1 to 500) do cancel out.</p>
<p>The first portion of the grand sum can be written as:
(-494 -481 -468 … -13 +0 +13 … +468 +481 +494) = 0</p>
<p>Which, after factoring, is equivalent to:
(13)(-38 -37 -36 … -1 +0 +1 +2 … +38) = 0</p>
<p>As previously noted, -500 and +500 don’t come into play since they are not multiples of 13.</p>
<p>
</p>
<p>The second portion of the grand sum looks like:
(507 + 520 +533 … +1482 + 1495)</p>
<p>which, after factoring, is equivalent to:
(13)( 39 + 40 + 41 + … + 114 + 115)</p>
<p>This sum equals the product of the number of terms and the average term
( 39 + 40 + 41 + … + 114 + 115) = (115-39 + 1) (39+115) / 2</p>
<p>Multiply by 13
= (13) (115-39 + 1) (39+115) / 2
= (13) (77) (77)
= 77077 (the second portion of the grand sum)
Add to zero (the first portion of the grand sum)</p>
<p>Note that we could do the problem without factoring by taking the sum
(-494 -481 -468 … -13 +0 +13 …+1482 + 1495)</p>
<p>With 154 terms (1495 - -494 + 1) and an average term of 500.5
for calculated sum of 77,077</p>
<p>Sorry, the 154 terms should be (1495 - -494)/13 + 1 = 154</p>
<p>for calculated sum of </p>
<p>154 X 500.5 = 77,077</p>
<p>My$0.02 laid it out really well … an excellent explanation, I think.</p>