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Arithmetic Sequence ProblemsArithmetic Sequence Problems
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Question 1 of 3
1. Question
Find the sequence given that:`U_5+U_13=126``U_9+U_17=182`Hint
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General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, transform the `5th` and `13th` terms into general rule form5th Term$$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{5}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{5}-1)\color{#00880A}{d}]$$ Substitute known values `U_5` `=` `a+4d` Distribute 13th Term$$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{13}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{13}-1)\color{#00880A}{d}]$$ Substitute known values `U_13` `=` `a+12d` Distribute Next, add the first pair general forms`U_5+U_13` `=` `(a+4d)+(a+12d)` `126``divide2` `=` `2a+16d``divide2` Divide both sides by `2` `63` `=` `a+8d` Next, transform the `9th` and `17th` terms into general rule form9th Term$$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{9}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{9}-1)\color{#00880A}{d}]$$ Substitute known values `U_9` `=` `a+8d` Distribute 17th Term$$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{17}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{17}-1)\color{#00880A}{d}]$$ Substitute known values `U_17` `=` `a+16d` Distribute Next, add the second pair general forms`U_9+U_17` `=` `(a+4d)+(a+12d)` `182``divide2` `=` `2a+24d``divide2` Divide both sides by `2` `91` `=` `a+12d` Next, solve for the value of `d` by subtracting the first combined general rule form from the second combined general rule form`91``-``63` `=` `(``a+12d``)-(``a+8d``)` `28``divide4` `=` `4d``divide4` Divide both sides by `4` `7` `=` `d` `d` `=` `7` Next, substitute `d` to one of the combined general rule forms to solve for `a``63` `=` `a+8``d` `63` `=` `a+8(``7``)` Substitute `d=7` `63` `-56` `=` `a+56` `-56` Subtract `56` from both sides `7` `=` `a` `a` `=` `7` Finally, start with `a=7` and keep adding `d=7` to its value to get the sequence`U_1` `=` `7` `U_2` `=` `7+``7` `=` `14` `U_3` `=` `14+``7` `=` `21` `U_4` `=` `21+``7` `=` `28` `7+14+21+28…` `7+14+21+28…` -
Question 2 of 3
2. Question
The short end of a fence has a height of `140` cm. After about `71` pieces of timber, the height of the fence is `210` cm. Find:`(i)` The difference in height between each fence`(ii)` The total height of the fences in metres
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`(i)` `d=` (1)`\text(cm)``(ii)` `S_n=` (124.25)`\text(m)`
Hint
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$`(i)` Finding the difference in height between each fence (`d`)Substitute the known values to the general rule`\text(Number of terms)``[n]` `=` `71` `\text(First term)``[a]` `=` `140` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{71}}$$ `=` $$\color{#e65021}{140}+[(\color{#9a00c7}{71}-1)\color{#00880A}{d}]$$ Substitute known values `210` `-140` `=` `140+70d` `-140` Subtract `140` from both sides `70``divide70` `=` `70d``divide70` Divide both sides by `70` `1` `=` `d` `d` `=` `1 \text(cm)` `(ii)` Finding the total height of the fences in metres (`S_n`)Substitute the known values to the sum formula`\text(Number of terms) [n]` `=` `71` `\text(First Term) [a]` `=` `140` `\text(Common Difference) [d]` `=` `1` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$S_{\color{#9a00c7}{71}}$$ `=` $$\frac{\color{#9a00c7}{71}}{2}[2\cdot\color{#e65021}{140}+(\color{#9a00c7}{71}-1)\color{#00880A}{1}]$$ Substitute known values `=` $$35\frac{1}{2}(280+70)$$ Evaluate `=` $$35\frac{1}{2}(350)$$ `=` `12 425 \text(cm)` Finally, convert the centimetres into metres`1 \text(metre)` `=` `100 \text(centimetres)` `12425xx1/(100)` `=` `124.25 \text(cm)` `(i) d=1 \text(cm)``(ii) S_n=124.25 \text(m)` -
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Question 3 of 3
3. Question
A stack of cans has `1` can at the top and then each row has `2` more cans than the previous row. Find:`(i)` The number of cans in the `21st` row`(ii)` The row with `61` cans`(iii)` The number of rows if there are a total of `1296` cans
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`(i)` `U_21=` (41)`(ii)` `n=` (31)`(iii)` `n=` (36)
Hint
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$`(i)` Finding the number of cans in the `21st` row (`U_21`)Substitute the known values to the general rule`\text(Number of terms)``[n]` `=` `21` `\text(First term)``[a]` `=` `1` `\text(Common Difference)``[d]` `=` `2` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{21}}$$ `=` $$\color{#e65021}{1}+[(\color{#9a00c7}{21}-1)\color{#00880A}{2}]$$ Substitute known values `=` `1+(20)2` Evaluate `=` `1+40` `=` `41` `(ii)` Finding the row with `61` cans (`n`)Substitute the known values to the general rule and solve for `n``\text(Nth term)``[U_n]` `=` `61` `\text(First term)``[a]` `=` `1` `\text(Common Difference)``[d]` `=` `2` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$61$$ `=` $$\color{#e65021}{1}+[(\color{#9a00c7}{n}-1)\color{#00880A}{2}]$$ Substitute known values `61` `=` `1+(2n-2)` Distribute `61` `+1` `=` `-1+2n` `+1` Add `1` to both sides `62``divide2` `=` `2n``divide2` Divide both sides by `2` `31` `=` `n` `n` `=` `31` `(iii)` Finding the number of rows if `S_n=1296` (`n`)Substitute the known values to the sum formula and solve for `n``\text(Sum of terms)``[S_n]` `=` `1296` `\text(First Term) [a]` `=` `1` `\text(Common Difference) [d]` `=` `2` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$1296$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\cdot\color{#e65021}{1}+(\color{#9a00c7}{n}-1)\color{#00880A}{2}]$$ Substitute known values `1296` `=` $$\frac{n}{2}(2+2n-2)$$ Evaluate `1296``times2` `=` `[n/2(2n)]``times2` Multiply both sides by `2` `2592` `=` `n(2n)` `2/2=1` `2592``divide2` `=` `2n^2``divide2` Divide both sides by `2` `sqrt1296` `=` `sqrt(n^2)` Find the square root of both sides `36` `=` `n` `n` `=` `36` `(i) U_21=41``(ii) n=31``(iii) n=36` -