<p>I'd like to see how you guys would solve this: </p>
<p>* If the average of 18 consecutive odd integers is 534, what is the least of these integers? *</p>
<p>I solved it like this: </p>
<p>Sum of numbers: (534)(18)
Equation for sum of arith series:
sum = n/2[2t1 + (n-1)d]
9612 = 9[2t1 + (17)(2)]
t1 = 517</p>
<p>This is an SAT Reasoning problem, though. Are there any non "advanced" methods to do this problem?</p>
<p>well i used common sense to realize that because 534 is an even number and we are looking at odd numbers, then 533 is the 9th term(and 535 is the 10th). so i subtracted 8(2) from 533 because i had to go back 8 terms and the terms are 2 units apart. 517.</p>
<p>Think about it logically: If the average of the numbers is 534, and the numbers are consecutive, then 534 is also the median. (Alright, so maybe that’s more statistically than logically).</p>
<p>Basically, all you need to do is count outwards. If 534 is the middle, then the two middle odd terms are both 1 away from 534. (It’s more a visual thing than a math thing).</p>
<p>Start with 534
_ (534) _<br>
The blanks are 533 and 535, the two consecutive odd integers that surround 534. The two middle integers of a set of 18 terms are the 9th and 10th term. 533 is the 9th term.</p>
<p>The whole picture is:</p>
<p>_ _ _ _ _ _ _ _ 533 (534) 535 _ _ _ _ _ _ _ _ </p>
<p>Each blank is a consecutive odd integer, so count backwards by 2 from 533 until you get to the first term:</p>
<p>517 519 521 523 525 527 529 531 533 (534)
1----2—3–4—5—6—7—8—9</p>
<p>The answer is 517</p>
<p>^ great minds think alike Godfatherbob</p>
<p>Sweet. That’s a cool way to do it.</p>
<p>Arachnotron i would support your method.It is always good to know how to solve a problem by using pure algebra</p>
<p>i went
(1+2+3+4+5+6+7+8+9…+17)2 +18x = 534*18
18x = 9306
x=517</p>
<p>i used this way cuz it didnt require too much thinking and i can just punch all the numbers into a calculator quickly</p>
<p>JSMGH what if the problem says ‘‘60 consecutive integers’’ instead of 18 ::}}</p>
<p>An alternate (almost non-advanced) route.
The average (and the median) of the first 18 positive consecutive odd integers (1, 3, …, 35) is 18.
Let’s advance them all up the number line so that their median (and the average) becomes 534.
The shift is 534 - 18 = 516, so the least of the set becomes
1 + 516 = 517.</p>
<p>Edit.
A useful fact: the average of the first n positive consecutive odd integers is n.</p>
<p>if that was a multiple choice question i’d try every answer lol</p>
<p>i did what godfatherbob did.
it’s easy to see that 534 is the middle number,and the left and right balance out each other,which would still end up being 534. then just count…</p>
<p>I did it this way (I tend to think algebraically about this stuff):</p>
<p>x is the least integer, and every odd integer after that is 2 more than the previous, so we have:
x+(x+2)+(x+4)+(x+6)+…+(x+34) = (18)(534) = 9612
18x + 306 = 9612
18x = 9306
x = 517</p>
<p>Maybe not the fastest, but it works without knowing any advanced method.</p>