Question 1102223
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This will be pretty much the same answer as provided by the other tutor, but using informal logical analysis instead of formal mathematical formulas.<br>
Assuming that "between 1 and 200" includes 200, the sum is 5+10+...+195+200.<br>
The number of terms in the sum is 200/5 = 40.<br>
Since the numbers are equally spaced, the 40 numbers can be paired into 20 pairs, each with a sum of 205: 5+200 = 205; 10+195 = 205; 15+190 = 205; etc.<br>
So the sum of all the numbers is the sum of 20 pairs, with each pair having a sum of 205: 205*20 = 4100.<br>
And here is an alternative way to think about getting the final sum, once we have found that there are 40 terms in the sum.<br>
Again since the numbers are equally spaced, the average of all the terms is the average of the first and last terms: (5+200)/2 = 102.5<br>
Then the sum of all the numbers is that average, multiplied by the number of terms: 40(102.5) = 4100.<br>
To summarize these two different ways of finding the sum, we have either<br>
(a) (number of pairs)*(sum of first and last) = {{{(n/2)(a1+an)}}}
or
(b) (number of numbers)*(average of first and last) = {{{n((a1+an)/2)}}}<br>
Those two formulas, representing different ways of thinking of getting the answer, are clearly mathematically equivalent.