Geometric Series (Sum)
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Question 1 of 5
1. Question
Find the sum of the first `7` terms`96+48+24…`- `S_7=` (190.5)
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Sum of a Geometric Sequence
$$S_{\color{#9a00c7}{n}}=\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$$Common Ratio Formula
$$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$$First, solve for the value of `r`.$$\color{#00880A}{r}$$ `=` $$\frac{U_2}{U_1}$$ `=` $$\frac{48}{96}$$ Substitute the first and second term `=` `1/2` Next, substitute the known values to the formula`\text(Number of Terms) [n]` `=` `7` `\text(First term) [a]` `=` `96` `\text(Common Ratio) [r]` `=` `1/2` $$S_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$$ $$S_{\color{#9a00c7}{7}}$$ `=` $$\color{#e65021}{96}\left(\frac{1-\color{#00880A}{\frac{1}{2}}^{\color{#9a00c7}{7}}}{1-\color{#00880A}{\frac{1}{2}}}\right)$$ Substitute known values `=` `(96[1-(1/128)])/(1/2)` Evaluate `=` `(96(127/128))/(1/2)` `=` `192(127/128)` `=` `381/2` `=` `190.5` `S_7=190.5` -
Question 2 of 5
2. Question
Given that `S_n=42 3/4`, find the value of n`64-32+18-8…`- `n=` (9)
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Sum of a Geometric Sequence
$$S_{\color{#9a00c7}{n}}=\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$$Common Ratio Formula
$$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$$First, solve for the value of `r`.$$\color{#00880A}{r}$$ `=` $$\frac{U_2}{U_1}$$ `=` $$\frac{-32}{64}$$ Substitute the first and second term `=` `-1/2` Next, substitute the known values to the formula`\text(Sum of terms) [S_n]` `=` `42 3/4` `\text(First term) [a]` `=` `64` `\text(Common Ratio) [r]` `=` `-1/2` $$S_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$$ $$42\frac{3}{4}$$ `=` $$\color{#e65021}{64}\left(\frac{1-\color{#00880A}{-\frac{1}{2}}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{-\frac{1}{2}}}\right)$$ Substitute known values `(42 3/4)``times 3/2` `=` `[(64(1-(-1/2)^n))/(3/2)]``times 3/2` Multiply both sides by `3/2` `513/8``divide 64` `=` `[64(1-(-1/2)^n)]``divide 64` Divide both sides by `64` `513/512``times(-1)` `=` `1-(-1/2)^n``times(-1)` Multiply both sides by `(-1)` `-513/512` `+1` `=` `-1+(-1/2)^n` `+1` Add `1` to both sides `-1/512``times(-1)` `=` `(-1/2)^n``times(-1)` Multiply both sides by `(-1)` `1/(2^9)` `=` `1/(2^n)` `512=2^9` `9` `=` `n` Equate the exponents of the denominator `n` `=` `9` `n=9` -
Question 3 of 5
3. Question
Given the sequence `1+5+25+125+625…`, find:`(i)` The sum of the first `12` terms`(ii)` The number of terms needed to have `S_n=97656`-
`(i)` `U_12=` (61035156)`(ii)` `n=` (8)
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Sum of a Geometric Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$Common Ratio Formula
$$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$$`(i)` Finding the sum of the first `12` termsFirst, solve for the value of `r`.$$\color{#00880A}{r}$$ `=` $$\frac{U_2}{U_1}$$ `=` $$\frac{5}{1}$$ Substitute the first and second term `=` `5` Next, substitute the known values to the formula`\text(Number of Terms) [n]` `=` `12` `\text(First term) [a]` `=` `1` `\text(Common Ratio) [r]` `=` `5` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$ $$S_{\color{#9a00c7}{12}}$$ `=` $$\frac{\color{#e65021}{1}(\color{#00880A}{5}^{\color{#9a00c7}{12}}-1)}{\color{#00880A}{5}-1}$$ Substitute known values `=` `(5^12-1)/(4)` Evaluate `=` `(244 140 624)/4` `=` `61 035 156` `(ii)` Finding the number of terms needed to have `S_n=97656`Substitute the known values to the formula`\text(Sum of terms) [S_n]` `=` `97656` `\text(First term) [a]` `=` `1` `\text(Common Ratio) [r]` `=` `5` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$ $$97656$$ `=` $$\frac{\color{#e65021}{1}(\color{#00880A}{5}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{5}-1}$$ Substitute known values `97656``times4` `=` `(5^n-1)/4``times4` Multiply both sides by `4` `390 624` `+1` `=` `5^n-1` `+1` Add `1` to both sides `390 625` `=` `5^n` Use the `log` function in your calculator and solve for `n``log390 625` `=` `n log5` `log_b x^p=p log_b x` `log390 625``divide log5` `=` `n log5``divide log5` Divide both sides by `log5` `8` `=` `n` `n` `=` `8` `(i) U_12=61 035 156``(ii) n=8` -
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Question 4 of 5
4. Question
Find the value of `n` given that `S_n``>``5000``5+10+20…`- `n=` (10)
Hint
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Sum of a Geometric Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$Common Ratio Formula
$$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$$First, solve for the value of `r`.$$\color{#00880A}{r}$$ `=` $$\frac{U_2}{U_1}$$ `=` $$\frac{10}{5}$$ Substitute the first and second term `=` `2` Substitute the known values to the formula`\text(First term) [a]` `=` `5` `\text(Common Ratio) [r]` `=` `2` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$$ $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{5}(\color{#00880A}{2}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{2}-1}$$ Substitute known values `S_n` `=` `5(2^n-1)` Evaluate Substitute the value of `S_n` to the inequality`S_n` `>` `5000` `5(2^n-1)` `>` `5000` Substitute `S_n=5(2^n-1)` `5(2^n-1)``divide5` `>` `5000``divide5` Divide both sides by `5` `2^n-1` `+1` `>` `1000` `+1` Add `1` to both sides `2^n` `>` `1001` Use the `log` function in your calculator and solve for `n``n log2` `>` `log1001` `log_b x^p=p log_b x` `n log2``divide log2` `>` `log1001``divide log2` Divide both sides by `log2` `n` `>` `9.96722` `n` `=` `10` Rounded to a whole number `n=10` -
Question 5 of 5
5. Question
Find the value of `n` given that `S_n``>``49.99``40+8+8/5…`- `n=` (6)
Hint
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Incorrect
Sum of a Geometric Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#e65021}{a}(1-\color{#00880A}{r}^{\color{#9a00c7}{n}})}{1-\color{#00880A}{r}}$$Common Ratio Formula
$$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$$First, solve for the value of `r`.$$\color{#00880A}{r}$$ `=` $$\frac{U_2}{U_1}$$ `=` $$\frac{8}{40}$$ Substitute the first and second term `=` `1/5` Substitute the known values to the formula`\text(First term) [a]` `=` `40` `\text(Common Ratio) [r]` `=` `1/5` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{a}(1-\color{#00880A}{r}^{\color{#9a00c7}{n}})}{1-\color{#00880A}{r}}$$ $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#e65021}{40}[1-(\color{#00880A}{\frac{1}{5}})^{\color{#9a00c7}{n}}]}{1-\color{#00880A}{\frac{1}{5}}}$$ Substitute known values `=` `(40[1-(1/5)^n])/(4/5)` Evaluate `=` `50[1-(1/5)^n]` Substitute the value of `S_n` to the inequality`S_n` `>` `49.99` `50[1-(1/5)^n]` `>` `49.99` Substitute `S_n=50[1-(1/5)^n]` `50[1-(1/5)^n]``divide5` `>` `49.99``divide5` Divide both sides by `50` `1-(1/5)^n` `-1` `>` `(49.99)/50` `-1` Subtract `1` from both sides `-(1/5)^n``times(-1)` `>` `(49.99)/50-1``times(-1)` Multiply both sides by `-1` `(1/5)^n` `>` `1-(49.99)/50` `(1/5)^n` `>` `0.0002` Use the `log` function in your calculator and solve for `n``n log (1/5)` `>` `log0.0002` `log_b x^p=p log_b x` `n log (1/5)``divide log (1/5)` `>` `log0.0002``divide log (1/5)` Divide both sides by `log (1/5)` `n` `>` `5.29202` `n` `=` `6` Rounded to a whole number `n=6`