Precalculus: What is the value of this sigma notation?

In summary: 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
  • #1
ukumure
5
0
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!
 
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  • #2
Do you have definitions for \(\displaystyle f\) and \(\displaystyle g\)?
 
  • #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.
 
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  • #4
Greg said:
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!
 
  • #5
Greg said:
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

$g(i) = \dfrac{60}{21}$ ?
 
  • #6
skeeter said:
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 .
 
  • #7
skeeter said:
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 $$ \sum_{i=1}^{50} \frac{7 g(i)-f(i)+12}2 = \frac12\sum_{i=1}^{50} (7 g(i)-f(i)+12) = \frac12\left(7\sum_{i=1}^{50}g(i) - \sum_{i=1}^{50}f(i) + \sum_{i=1}^{50}12\right) = \frac12(7*60 - 90 + 600) = 465.$$
 
  • #8
Opalg said:
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 ...
 
  • #9
skeeter said:
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.
 
  • #10
skeeter said:
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
 
  • #11
Aaargh!
 

Related to Precalculus: What is the value of this sigma notation?

1. What is the purpose of sigma notation in precalculus?

Sigma notation is a shorthand way of writing a series or sum of terms in mathematics. It allows us to easily represent and manipulate large sums without having to write out each individual term. In precalculus, sigma notation is often used to represent sequences or series of numbers.

2. How do you read and interpret sigma notation?

Sigma notation is read as "the sum of" or "the series of." The value below the sigma symbol represents the starting value of the index, while the value above the sigma symbol represents the ending value. The expression after the sigma symbol represents the terms of the series. For example, the sigma notation for the sum of the first 5 even numbers would be written as Σ(i = 1 to 5) 2i.

3. What is the difference between an upper and lower limit in sigma notation?

The upper limit in sigma notation represents the last term in the series, while the lower limit represents the first term. The index variable, usually denoted by i, starts at the lower limit and increases by 1 for each subsequent term until it reaches the upper limit.

4. How do you evaluate a series written in sigma notation?

To evaluate a series written in sigma notation, you can substitute the values of the index variable into the expression after the sigma symbol and then add up all the resulting terms. For example, to evaluate the series Σ(i = 1 to 5) 2i, you would substitute i = 1, 2, 3, 4, and 5 into the expression 2i and then add up the resulting terms (2 + 4 + 6 + 8 + 10 = 30).

5. What are some common uses of sigma notation in precalculus?

Sigma notation is commonly used in precalculus to represent arithmetic and geometric sequences, as well as series such as the sum of powers, the sum of factorials, and the sum of binomial coefficients. It is also used to represent infinite series, which are important in calculus and other areas of mathematics.

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