Geometric Sequence Problems

Question 1 of 6

Find the sequence given that:

$U_2+U_3=120$

$U_4+U_5=1920$

1. $6,12,24...$
2. $6,24,96...$
3. $6,18,54...$
4. $6,36,216...$

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General Rule of a Geometric Sequence

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$

First, transform the

$2nd$ and

$3rd$ terms into general rule form

2nd Term

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{2}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{2}-1}$	Substitute known values
$U_2$	$=$	$ar$	Evaluate

3rd Term

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{3}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{3}-1}$	Substitute known values
$U_2$	$=$	$ar^2$	Evaluate

Next, add the first pair of general forms

$U_2+U_3$	$=$	$(ar)+(ar^2)$
$120$	$=$	$ar(1+r)$	Factorise

Next, transform the

$4th$ and

$5th$ terms into general rule form

4th Term

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{4}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{4}-1}$	Substitute known values
$U_4$	$=$	$ar^3$	Evaluate

5th Term

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{5}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{5}-1}$	Substitute known values
$U_5$	$=$	$ar^4$	Evaluate

Next, add the second pair of general forms

$U_4+U_5$	$=$	$(ar^3)+(ar^4)$
$1920$	$=$	$ar^3(1+r)$	Factorise

Next, solve for the value of $r$ by dividing the second combined general rule form by the first combined general rule form

$1920$ $divide$ $120$	$=$	$($ $ar^3(1+r)$ $)divide($ $ar(1+r)$ $)$
$sqrt16$	$=$	$sqrt(r^2)$	Get the square root of both sides
$4$	$=$	$r$
$r$	$=$	$4$

Next, substitute $r$ to one of the combined general rule forms to solve for $a$

$120$	$=$	$a\color{#00880A}{r}(1+\color{#00880A}{r})$
$120$	$=$	$a($ $4$ $)(1+$ $4$ $)$	Substitute $r=4$
$120$	$=$	$4a(5)$
$120$ $divide20$	$=$	$20a$ $divide20$	Divide both sides by $20$
$6$	$=$	$a$
$a$	$=$	$6$

Finally, start with $a=6$ and keep multiplying by $r=4$ to its value to get the sequence

$U_1$	$=$	$6$
$U_2$	$=$	$6times$ $4$	$=$	$24$
$U_3$	$=$	$24times$ $4$	$=$	$96$

$6,24,96…$

Question 2 of 6

Find the sequence given that:

$U_2=-72$

$U_5=4608$

1. $18,-72,288,-1156,4608...$
2. $8,-72,648,-1156,4608...$
3. $12,-72,432,-2592,4608...$
4. $4,-72,648,-2592,4608...$

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General Rule of a Geometric Sequence

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$

First, transform the

$2nd$ and

$5th$ terms into general rule form

2nd Term

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{2}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{2}-1}$	Substitute known values
$-72$	$=$	$ar$	Evaluate

5th Term

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{5}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{5}-1}$	Substitute known values
$4608$	$=$	$ar^4$	Evaluate

Next, solve for the value of $r$ by dividing the $5th$ term’s general rule form by the $2nd$ term’s general rule form

$4608$ $divide($ $-72$ $)$	$=$	$($ $ar^4$ $)divide($ $ar$ $)$
$\sqrt[3]{-64}$	$=$	$\sqrt[3]{r^3}$	Get the cube root of both sides
$-4$	$=$	$r$
$r$	$=$	$-4$

Next, substitute $r$ to one of the combined general rule forms to solve for $a$

$a\color{#00880A}{r}$	$=$	$-72$
$a(\color{#00880A}{-4})$	$=$	$-72$	Substitute $r=-4$
$-4a$ $divide(-4)$	$=$	$-72$ $divide(-4)$	Divide both sides by $-4$
$a$	$=$	$18$

Finally, start with $a=18$ and keep multiplying by $r=-4$ to its value to get the sequence

$U_1$	$=$	$18$
$U_2$	$=$	$18times($ $-4$ $)$	$=$	$-72$
$U_3$	$=$	$-72times($ $-4$ $)$	$=$	$288$
$U_4$	$=$	$288times($ $-4$ $)$	$=$	$-1156$
$U_5$	$=$	$-1156times($ $-4$ $)$	$=$	$4608$

$18,-72,288,-1156,4608…$

Question 3 of 6

A photocopier’s zoom function magnifies an image by

$1.3$ times with each zoom. If the image of a tree is originally

$10 \text(mm)$ , what would be its size after zooming in

$12$ times?

Round your answer to a whole number

$U_12=$ (179) $\text(mm)$

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General Rule of a Geometric Sequence

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$

Substitute the known values to the general rule

$\text(Number of terms)$ $[n]$	$=$	$12$
$\text(First term)$ $[a]$	$=$	$10$
$\text(Common Ratio)$ $[r]$	$=$	$1.3$

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$U_{\color{#9a00c7}{12}}$	$=$	$\color{#e65021}{10}\cdot\color{#00880A}{1.3}^{\color{#9a00c7}{12}-1}$	Substitute known values
	$=$	$10\cdot1.3^{11}$	Evaluate
	$=$	$179.216$
	$=$	$179$	Rounded to a whole number

$U_12=179 \text(mm)$

Question 4 of 6

A photocopier’s zoom function magnifies an image by

$1.3$ times with each zoom. How many times should the zoom function be used to have the image’s size magnified to greater than

$500 \text(mm)$ ?

Round your answer to a whole number

(15) $\text(times)$

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General Rule of a Geometric Sequence

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$

First, substitute the known values to the general rule

$\text(Nth term)$ $[U_n]$	$=$	$500$
$\text(First term)$ $[a]$	$=$	$10$
$\text(Common Ratio)$ $[r]$	$=$	$1.3$

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$
$\color{#e65021}{10}\cdot\color{#00880A}{1.3}^{\color{#9a00c7}{n}-1}$	$=$	$500$	Substitute known values
${10(1.3)^(n-1)}$ $divide10$*	$=$	$500$ $divide10$	Divide both sides by $10$
$(1.3)^(n-1)$	$=$	$50$

Next, use the

$log$ function in your calculator and solve for $n$

$(n-1)log(1.3)$	$>$	$log50$	$log_b x^p=p log_b x$
$(n-1)log(1.3)$ $divide log(1.3)$	$>$	$log50$ $divide log(1.3)$	Divide both sides by $log(1.3)$

$n-1$ $+1$	$>$	$(log50)/(log(1.3))$ $+1$	Add $1$ to both sides

$n$	$>$	$15.910$
$n$	$=$	$16$	Rounded to a whole number

Finally, since we are asked how many times the zoom button should be pressed, we must not count in the first term.

Therefore, the button should be pressed

$15$ times.

$15 \text(times)$

Question 5 of 6

Joe goes fishing and comes back with

$8$ tons of fish the first week,

$4$ tons the second week and

$2$ tons the third week. How many tons would he be getting by the

$10th$ week?

Write fractions as “a/b”

$U_10=$ (1/64) $\text(tons)$

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General Rule of a Geometric Sequence

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$

Common Ratio Formula

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

First, solve for the value of $r$ .

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

	$=$	$\frac{4}{8}$	Substitute the first and second term

	$=$	$1/2$

Next, substitute the known values to the formula

$\text(Number of terms)$ $[n]$	$=$	$10$
$\text(First term)$ $[a]$	$=$	$8$

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

$U_{\color{#9a00c7}{n}}$	$=$	$\color{#e65021}{a}\color{#00880A}{r}^{\color{#9a00c7}{n}-1}$

$U_{\color{#9a00c7}{10}}$	$=$	$\color{#e65021}{10}\cdot\left(\color{#00880A}{\frac{1}{2}}\right)^{\color{#9a00c7}{10}-1}$	Substitute known values

	$=$	$8\cdot\left(\frac{1}{2}\right)^9$	Evaluate

	$=$	$8\cdot\frac{1}{512}$

	$=$	$1/64$

$U_10=1/64 \text(tons)$

Question 6 of 6

Joe goes fishing and comes back with

$8$ tons of fish the first week,

$4$ tons the second week and

$2$ tons the third week. How many tons in TOTAL would he be bringing back if he goes fishing for

$10$ weeks?

Round your answer to three decimal poaces

$S_10=$ (15.984) $\text(tons)$

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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 $r$ .

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

	$=$	$\frac{4}{8}$	Substitute the first and second term

	$=$	$1/2$

Next, substitute the known values to the formula

$\text(Number of terms)$ $[n]$	$=$	$10$
$\text(First term) [a]$	$=$	$8$

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

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

$S_{\color{#9a00c7}{10}}$	$=$	$\color{#e65021}{8}\left(\frac{1-\color{#00880A}{\frac{1}{2}}^{\color{#9a00c7}{10}}}{1-\color{#00880A}{\frac{1}{2}}}\right)$	Substitute known values

	$=$	${8}\left(\frac{1-\frac{1}{1024}}{\frac{1}{2}}\right)$	Evaluate

	$=$	$16[1-(1/(1024))]$

	$=$	$16((1023)/(1024))$

	$=$	$1023/64$	$16/1024=1/64$

	$=$	$15.984$	Rounded to three decimal places

$S_10=15.984 \text(tons)$

Geometric Sequence Problems

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General Rule of a Geometric Sequence

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General Rule of a Geometric Sequence

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Common Ratio Formula

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Sum of a Geometric Sequence

Common Ratio Formula

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