Sigma Notation (Summation)
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Question 1 of 4
1. Question
Evaluate$$\sum_{n=5}^{16}\;(5n-7)$$- (546)
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[\color{#e65021}{a}+\color{#00880A}{l}]$$First, substitute the values `x=5 \text(to) 16` to the expression$$\sum_{\color{#007DDC}{x=5}}^{\color{#007DDC}{16}}\;(5\color{#9a00c7}{n}-7)$$ `=` `(5(``5``)-7)+(5(``6``)-7)+(5(``7``)-7)…(5(``16``)-7)` `=` `18``+23+28…``73` The first term (`a`) is `18` and the last term (`l`) is `73`.Next, find the number of terms by subtracting the lower value of `x` from its highest value then adding `1`.`n` `=` `(16-5)+1` `=` `12` Finally, substitute the known values to the formula`n` `=` `12` `a` `=` `18` `l` `=` `73` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[\color{#e65021}{a}+\color{#00880A}{l}]$$ $$S_{\color{#9a00c7}{12}}$$ `=` $$\frac{\color{#9a00c7}{12}}{2}[\color{#e65021}{18}+\color{#00880A}{73}]$$ Substitute known values `=` `6[91]` Evaluate `=` `546` `S_(12)=546` -
Question 2 of 4
2. Question
Evaluate$$\sum_{x=4}^{10}\;3^n$$- (88533)
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Sum of a Geometric Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$First, substitute the values `x=4 \text(to) 10` to the expression$$\sum_{\color{#007DDC}{x=4}}^{\color{#007DDC}{10}}\;3^{\color{#007DDC}{n}}$$ `=` $$3^{\color{#007DDC}{4}}+3^{\color{#007DDC}{5}}+3^{\color{#007DDC}{6}}…3^{\color{#007DDC}{10}}$$ `=` `81``+243+729…59 049` The first term (`a`) is `81`.Next, find the common ratio (`r`) by dividing two consecutive terms.`r` `=` `3^5divide3^4` `=` `3^(5-4)` `=` `3` Next, find the number of terms by subtracting the lower value of `x` from its highest value then adding `1`.`n` `=` `(10-4)+1` `=` `7` Finally, substitute the known values to the formula`n` `=` `7` `a` `=` `81` `r` `=` `3` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$ $$S_{\color{#9a00c7}{7}}$$ `=` $$\frac{\color{#e65021}{81}(\color{#00880A}{3}^{\color{#9a00c7}{7}}-1)}{\color{#00880A}{3}-1}$$ Substitute known values `=` `(81(2186))/2` Evaluate `=` `(177 066)/2` `=` `88 533` `S_(7)=88 533` -
Question 3 of 4
3. Question
Find the limiting sum$$\sum_{r=2}^{∞}\;\left(\frac{4}{5}\right)^{r-1}$$- `S_∞=` (4)
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Limiting Sum Formula
$$\color{#9a00c7}{S_∞}=\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$$`\text(where) -1``<``r``<``1`Common Ratio Formula
$$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$$First, substitute the values `r=2..` to the expression$$\sum_{\color{#007DDC}{r=2}}^{∞}\;\left(\frac{4}{5}\right)^{r-1}$$ `=` $$\left(\frac{4}{5}\right)^{2-1}+\left(\frac{4}{5}\right)^{3-1}\left(\frac{4}{5}\right)^{4-1}…$$ `=` `4/5``+16/25+64/125…` The first term (`a`) is `4/5`.Next, find the common ratio (`r`) by dividing two consecutive terms.`r` `=` `16/25divide4/5` `=` `80/100` `=` `4/5` Simplify Finally, substitute the known values to the limiting sum formula`a` `=` `4/5` `r` `=` `4/5` $$\color{#9a00c7}{S_∞}$$ `=` $$\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$$ $$\color{#9a00c7}{S_∞}$$ `=` $$\frac{\color{#e65021}{\frac{4}{5}}}{1-\color{#00880A}{\frac{4}{5}}}$$ Substitute known values `=` `(4/5)/(1/5)` Evaluate `=` `20/5` `=` `4` `S_∞=4` -
Question 4 of 4
4. Question
Evaluate$$\sum_{r=3}^{6}\;(-1)^r\;r^2$$- (18)
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A sigma notation can be used to get the sum of values without having a common difference or ratio.Substitute the values `r=3 \text(to) 6` to the expression and evaluate$$\sum_{\color{#007DDC}{r=3}}^{\color{#007DDC}{6}}\;(-1)^{\color{#9a00c7}{r}}\;\color{#9a00c7}{r}^2$$ `=` $$\left[(-1)^{\color{#007DDC}{3}}\cdot \color{#007DDC}{3}^2\right]+\left[(-1)^{\color{#007DDC}{4}}\cdot \color{#007DDC}{4}^2\right]+\left[(-1)^{\color{#007DDC}{5}}\cdot \color{#007DDC}{5}^2\right]+\left[(-1)^{\color{#007DDC}{6}}\cdot \color{#007DDC}{6}^2\right]$$ `=` `[(-1)*9]+[1*16]+[(-1)*25]+[1*36]` Evaluate `=` `-9+16-25+36` `=` `18` `S_n=18`