How can I find the sum of numbers divisible by 3 that are between 1 and 1000 if I don't know how many elements there are in the arithmetic progression?
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2$$a_1=3;,;;a_n=999;,;;d=3$$ Find now what is $;n;$ and then do the arith. progression's sum. – Timbuc Jan 22 '15 at 16:31
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You can't. So find out (as per @Timbuc comment). – Joffan Jan 22 '15 at 16:32
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1How many multiples of three are there between $1$ and $1000$? Well, they range from $3\cdot1=3$ to $3\cdot333=999$… – Akiva Weinberger Jan 22 '15 at 16:32
2 Answers
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For guidance, I'll work a general problem:
How can I find the sum of numbers divisible by 17 between 432 and 6789?
I need to find the first element, the last element and the number of elements.
The first element $a_S =17 \left\lceil \frac{432}{17} \right\rceil = 17\times 26 = 442$
The last element $a_E = 17 \left\lfloor \frac{6789}{17} \right\rfloor = 17\times 399 = 6783$
Number of elements $N= 399-26+1 = 374$
Sum: $$S=\frac{N(a_S+a_E)}{2}=\frac{374(442+6783)}{2} = 13521075$$
Note: $\lceil x\rceil$ means round $x$ up to the nearest integer, and $\lfloor x\rfloor$ means round down.
Joffan
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Well you can easily find the number of elements. Consider that three is the common difference, and the starting number is 3 (first number divisible by three in the given range). Find the maximum value of $n$ for which we are still within the range. Then you can apply the formula for finding the sum of an arithmetic progression with $n$ terms, first term 3 and common difference as 3.
Gummy bears
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@Kimball Either way of course. But I believe to actually earn the basics of sequences and progression, it's important to learn the actual method too. Then you can apply shortcuts. :) – Gummy bears Jan 22 '15 at 16:44