Geometric Series (Sum)

Question 1 of 5

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

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

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

$12$ terms

First, 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 $log5$
$8$	$=$	$n$
$n$	$=$	$8$

$(i) U_12=61 035 156$

$(ii) n=8$

Question 4 of 5

Find the value of

$n$ given that

$S_n$

$>$

$5000$

$5+10+20…$

$n=$ (10)

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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

Find the value of

$n$ given that

$S_n$

$>$

$49.99$

$40+8+8/5…$

$n=$ (6)

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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

$n=6$

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Quiz summary

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1. Question

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Great Work!

Sum of a Geometric Sequence

Common Ratio Formula

2. Question

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Correct!

Sum of a Geometric Sequence

Common Ratio Formula

3. Question

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Keep Going!

Sum of a Geometric Sequence

Common Ratio Formula

4. Question

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Fantastic!

Sum of a Geometric Sequence

Common Ratio Formula

5. Question

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Excellent!

Sum of a Geometric Sequence

Common Ratio Formula

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