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Question 172794: Find the sum

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Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Note: there are 305-150%2B1=156 terms that need to be added.


sum%284i-5%2Ci=150%2C305%29 Start with the given summation


sum%284i%2Ci=150%2C305%29%2Bsum%28-5%2Ci=150%2C305%29 Break up the summation


sum%284i%2Ci=150%2C305%29%2B%28-5%29%2A%28156%29 Since the 2nd term has no variables, we can simply multiply the inner value -5 by the number of terms 156


sum%284i%2Ci=150%2C305%29-780 Multiply


4%2Asum%28i%2Ci=150%2C305%29-780 Pull out the coefficient 4



Now this is where the formula sum%28i%2Ci=1%2Cn%29=%28n%28n%2B1%29%29%2F2 comes in. There's one small problem though: the terms NEED to start at 1. So what this means is that we have to break sum%28i%2Ci=150%2C305%29 into


sum%28i%2Ci=150%2C305%29=sum%28i%2Ci=1%2C305%29-sum%28i%2Ci=1%2C149%29


sum%28i%2Ci=150%2C305%29=%28305%28305%2B1%29%29%2F2-%28149%28149%2B1%29%29%2F2 Now use the formula sum%28i%2Ci=1%2Cn%29=%28n%28n%2B1%29%29%2F2


sum%28i%2Ci=150%2C305%29=%28305%28306%29%29%2F2-%28149%28150%29%29%2F2 Add


sum%28i%2Ci=150%2C305%29=305%28153%29-149%2875%29 Reduce


sum%28i%2Ci=150%2C305%29=46665-11175 Multiply


sum%28i%2Ci=150%2C305%29=35490 Subtract


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So the expression

4%2Asum%28i%2Ci=150%2C305%29-780


then becomes


4%2A%2835490%29-780


141960-780 Multiply


141180 Subtract


So this means that sum%284i-5%2Ci=150%2C305%29=141180