Geometric Sequence Problems
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Question 1 of 6
1. Question
Find the sequence given that:`U_2+U_3=120``U_4+U_5=1920`Hint
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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 form2nd 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 form4th 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…` `6,24,96…` -
Question 2 of 6
2. Question
Find the sequence given that:`U_2=-72``U_5=4608`Hint
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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 form2nd 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…` `18,-72,288,-1156,4608…` -
Question 3 of 6
3. Question
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
4. Question
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)`
Hint
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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
5. Question
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
6. Question
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)`