Geometric Series (Sum)

Question 1 of 5

Find the sum of the first

7

$7$ terms

96 + 48 + 24 \dots

$96+48+24…$

$S_{7} =$ $S_7=$ (190.5)

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Sum of a Geometric Sequence

S_{n} = a (\frac{1 - r^{n}}{1 - r})

$S_{\color{#9a00c7}{n}}=\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$

Common Ratio Formula

r = \frac{U_{2}}{U_{1}} = \frac{U_{3}}{U_{2}}

$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$

First, solve for the value of $r$ $r$ .

$\color{#00880A}{r}$	$=$ $=$	$\frac{U_2}{U_1}$

	$=$ $=$	$\frac{48}{96}$	Substitute the first and second term

	$=$ $=$	$\frac{1}{2}$ $1/2$

Next, substitute the known values to the formula

$Number of Terms [n]$ $\text(Number of Terms) [n]$	$=$ $=$	$7$ $7$

$First term [a]$ $\text(First term) [a]$	$=$ $=$	$96$ $96$

$Common Ratio [r]$ $\text(Common Ratio) [r]$	$=$ $=$	$\frac{1}{2}$ $1/2$

$S_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$

$S_{\color{#9a00c7}{7}}$	$=$ $=$	$\color{#e65021}{96}\left(\frac{1-\color{#00880A}{\frac{1}{2}}^{\color{#9a00c7}{7}}}{1-\color{#00880A}{\frac{1}{2}}}\right)$	Substitute known values

	$=$ $=$	$\frac{96 [1 - (\frac{1}{128})]}{\frac{1}{2}}$ $(96[1-(1/128)])/(1/2)$	Evaluate

	$=$ $=$	$\frac{96 (\frac{127}{128})}{\frac{1}{2}}$ $(96(127/128))/(1/2)$

	$=$ $=$	$192 (\frac{127}{128})$ $192(127/128)$

	$=$ $=$	$\frac{381}{2}$ $381/2$

	$=$ $=$	$190.5$ $190.5$

S_{7} = 190.5

$S_7=190.5$

Question 2 of 5

Given that

S_{n} = 42 \frac{3}{4}

$S_n=42 3/4$ , find the value of n

64 - 32 + 18 - 8 \dots

$64-32+18-8…$

$n =$ $n=$ (9)

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Sum of a Geometric Sequence

S_{n} = a (\frac{1 - r^{n}}{1 - r})

$S_{\color{#9a00c7}{n}}=\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$

Common Ratio Formula

r = \frac{U_{2}}{U_{1}} = \frac{U_{3}}{U_{2}}

$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$

First, solve for the value of $r$ $r$ .

$\color{#00880A}{r}$	$=$ $=$	$\frac{U_2}{U_1}$

	$=$ $=$	$\frac{-32}{64}$	Substitute the first and second term

	$=$ $=$	$- \frac{1}{2}$ $-1/2$

Next, substitute the known values to the formula

$Sum of terms [S_{n}]$ $\text(Sum of terms) [S_n]$	$=$ $=$	$42 \frac{3}{4}$ $42 3/4$

$First term [a]$ $\text(First term) [a]$	$=$ $=$	$64$ $64$

$Common Ratio [r]$ $\text(Common Ratio) [r]$	$=$ $=$	$- \frac{1}{2}$ $-1/2$

$S_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}\left(\frac{1-\color{#00880A}{r}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{r}}\right)$

$42\frac{3}{4}$	$=$ $=$	$\color{#e65021}{64}\left(\frac{1-\color{#00880A}{-\frac{1}{2}}^{\color{#9a00c7}{n}}}{1-\color{#00880A}{-\frac{1}{2}}}\right)$	Substitute known values

$(42 \frac{3}{4})$ $(42 3/4)$ $\times \frac{3}{2}$ $times 3/2$	$=$ $=$	$⎡ ⎢ ⎣ \frac{64 (1 - {(- \frac{1}{2})}^{n})}{\frac{3}{2}} ⎤ ⎥ ⎦$ $[(64(1-(-1/2)^n))/(3/2)]$ $\times \frac{3}{2}$ $times 3/2$	Multiply both sides by $\frac{3}{2}$ $3/2$

$\frac{513}{8}$ $513/8$ $\div 64$ $divide 64$	$=$ $=$	$[64 (1 - {(- \frac{1}{2})}^{n})]$ $[64(1-(-1/2)^n)]$ $\div 64$ $divide 64$	Divide both sides by $64$ $64$

$\frac{513}{512}$ $513/512$ $\times (- 1)$ $times(-1)$	$=$ $=$	$1 - {(- \frac{1}{2})}^{n}$ $1-(-1/2)^n$ $\times (- 1)$ $times(-1)$	Multiply both sides by $(- 1)$ $(-1)$

$- \frac{513}{512}$ $-513/512$ $+ 1$ $+1$	$=$ $=$	$- 1 + {(- \frac{1}{2})}^{n}$ $-1+(-1/2)^n$ $+ 1$ $+1$	Add $1$ $1$ to both sides

$- \frac{1}{512}$ $-1/512$ $\times (- 1)$ $times(-1)$	$=$ $=$	${(- \frac{1}{2})}^{n}$ $(-1/2)^n$ $\times (- 1)$ $times(-1)$	Multiply both sides by $(- 1)$ $(-1)$

$\frac{1}{2^{9}}$ $1/(2^9)$	$=$ $=$	$\frac{1}{2^{n}}$ $1/(2^n)$	$512 = 2^{9}$ $512=2^9$

$9$ $9$	$=$ $=$	$n$ $n$	Equate the exponents of the denominator
$n$ $n$	$=$ $=$	$9$ $9$

n = 9

$n=9$

Question 3 of 5

Given the sequence

1 + 5 + 25 + 125 + 625 \dots

$1+5+25+125+625…$ , find:

(i)

$(i)$ The sum of the first

12

$12$ terms

(i i)

$(ii)$ The number of terms needed to have

S_{n} = 97656

$S_n=97656$

$(i)$ $(i)$ $U_{12} =$ $U_12=$ (61035156)

$(i i)$ $(ii)$ $n =$ $n=$ (8)

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Sum of a Geometric Sequence

S_{n} = \frac{a (r^{n} - 1)}{r - 1}

$S_{\color{#9a00c7}{n}}=\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$

Common Ratio Formula

r = \frac{U_{2}}{U_{1}} = \frac{U_{3}}{U_{2}}

$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$

(i)

$(i)$ Finding the sum of the first

12

$12$ terms

First, solve for the value of $r$ $r$ .

$\color{#00880A}{r}$	$=$ $=$	$\frac{U_2}{U_1}$

	$=$ $=$	$\frac{5}{1}$	Substitute the first and second term

	$=$ $=$	$5$ $5$

Next, substitute the known values to the formula

$Number of Terms [n]$ $\text(Number of Terms) [n]$	$=$ $=$	$12$ $12$
$First term [a]$ $\text(First term) [a]$	$=$ $=$	$1$ $1$
$Common Ratio [r]$ $\text(Common Ratio) [r]$	$=$ $=$	$5$ $5$

$S_{\color{#9a00c7}{n}}$	$=$ $=$	$\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$

$S_{\color{#9a00c7}{12}}$	$=$ $=$	$\frac{\color{#e65021}{1}(\color{#00880A}{5}^{\color{#9a00c7}{12}}-1)}{\color{#00880A}{5}-1}$	Substitute known values

	$=$ $=$	$\frac{5^{12} - 1}{4}$ $(5^12-1)/(4)$	Evaluate

	$=$ $=$	$\frac{244140624}{4}$ $(244 140 624)/4$

	$=$ $=$	$61035156$ $61 035 156$

(i i)

$(ii)$ Finding the number of terms needed to have

S_{n} = 97656

$S_n=97656$

Substitute the known values to the formula

$Sum of terms [S_{n}]$ $\text(Sum of terms) [S_n]$	$=$ $=$	$97656$ $97656$
$First term [a]$ $\text(First term) [a]$	$=$ $=$	$1$ $1$
$Common Ratio [r]$ $\text(Common Ratio) [r]$	$=$ $=$	$5$ $5$

$S_{\color{#9a00c7}{n}}$	$=$ $=$	$\frac{\color{#e65021}{a}(\color{#00880A}{r}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{r}-1}$

$97656$	$=$ $=$	$\frac{\color{#e65021}{1}(\color{#00880A}{5}^{\color{#9a00c7}{n}}-1)}{\color{#00880A}{5}-1}$	Substitute known values

$97656$ $97656$ $\times 4$ $times4$	$=$ $=$	$\frac{5^{n} - 1}{4}$ $(5^n-1)/4$ $\times 4$ $times4$	Multiply both sides by $4$ $4$

$390624$ $390 624$ $+ 1$ $+1$	$=$ $=$	$5^{n} - 1$ $5^n-1$ $+ 1$ $+1$	Add $1$ $1$ to both sides
$390625$ $390 625$	$=$ $=$	$5^n$

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

$n=6$

Geometric Series (Sum)

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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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