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Precalculus: What is the value of this sigma notation?

ukumure

New member
Jan 11, 2021
3
Hi, I'm currently a Grade 11 student and I need help for this question (Precalculus):

If $\sum\limits_{i=1}^{50} f(i)=90$ and $\sum\limits_{i=30}^{50} g(i)=60$, what is the value of $\sum\limits_{i=1}^{50} (7 g(i)-f(i)+12)/(2)$?

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P.S. To those who could answer this, it would be a great help for me! Thank you so much!
 
Last edited by a moderator:

Greg

Perseverance
Staff member
Feb 5, 2013
1,391
Do you have definitions for \(\displaystyle f\) and \(\displaystyle g\)?
 

Greg

Perseverance
Staff member
Feb 5, 2013
1,391
Oh! Those definitions are implied... :poop:

\(\displaystyle f(i)=1.8.\) and \(\displaystyle g(i)=1.2\) both suffice as definitions for \(\displaystyle f,\,g\) if I am not mistaken... After a few basic calculations we may arrive at:

\(\displaystyle \frac{420-90+600}{2}=465\). Do you see that too?

Hint: use the fact that summation is associative and sum each addend separately with all operations being applied according to BEDMAS. Brackets around the numerator are omitted but in accordance with notational convention they are implied. The string

\(\displaystyle (7g(i)-f(i)+ 12)/2\) may be more useful to you.
 
Last edited:

ukumure

New member
Jan 11, 2021
3
Oh! Those definitions are implied... :poop:

\(\displaystyle f(i)=1.8.\) and \(\displaystyle g(i)=1.2\) both suffice as definitions for \(\displaystyle f,\,g\) if I am not mistaken... After a few basic calculations we may arrive at:

\(\displaystyle \frac{420-90+600}{2}=465\). Do you see that too?

Hint: use the fact that summation is associative and sum each addend separately with all operations being applied according to BEDMAS. Brackets around the numerator are omitted but in accordance with notational convention they are implied. The string

\(\displaystyle (7g(i)-f(i)+ 12)/2\) may be more useful to you.
Thank you so much for helping me! ♥♥♥ This means a lot to me! Thank you!
 

skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
765
Oh! Those definitions are implied... :poop:

\(\displaystyle f(i)=1.8.\) and \(\displaystyle g(i)=1.2\) both suffice as definitions for \(\displaystyle f,\,g\) ...
If the assumption that both $f(i)$ and $g(i)$ are constants is correct, wouldn’t

$g(i) = \dfrac{60}{21}$ ?
 

Greg

Perseverance
Staff member
Feb 5, 2013
1,391
If the assumption that both $f(i)$ and $g(i)$ are constants is correct, wouldn’t

$g(i) = \dfrac{60}{21}$ ?
Yes, I agree. My error was missing \(\displaystyle i\) = 30 and assuming \(\displaystyle i\) = 1 .
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,740
If the assumption that both $f(i)$ and $g(i)$ are constants is correct, wouldn’t

$g(i) = \dfrac{60}{21}$ ?
There is no need to assume that $f(i)$ and $g(i)$ are constants. You just need to use the fact that $$ \sum_{i=1}^{50} \frac{7 g(i)-f(i)+12}2 = \frac12\sum_{i=1}^{50} (7 g(i)-f(i)+12) = \frac12\left(7\sum_{i=1}^{50}g(i) - \sum_{i=1}^{50}f(i) + \sum_{i=1}^{50}12\right) = \frac12(7*60 - 90 + 600) = 465.$$
 

skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
765
There is no need to assume that $f(i)$ and $g(i)$ are constants. You just need to use the fact that $$ \sum_{i=1}^{50} \frac{7 g(i)-f(i)+12}2 = \frac12\sum_{i=1}^{50} (7 g(i)-f(i)+12) = \frac12\left(7\sum_{i=1}^{50}g(i) - \sum_{i=1}^{50}f(i) + \sum_{i=1}^{50}12\right) = \frac12(7*60 - 90 + 600) = 465.$$
take another look at the indices for g(i) in the original post ...
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,740
take another look at the indices for g(i) in the original post ...
I should have looked more closely! As stated, the problem can have no definite solution.
 

ukumure

New member
Jan 11, 2021
3
take another look at the indices for g(i) in the original post ...
I got confused at the problem, especially the given i=30. But the instructor told me it was a typographical error. Thank you so much for your help! <3