Arithmetic Sequence Problems

Question 1 of 3

Find the sequence given that:

$U_5+U_13=126$

$U_9+U_17=182$

1. $7+14+21+28...$
2. $7+12+17+22...$
3. $7+11+15+19...$
4. $7+15+23+31...$

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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 form

5th 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 form

9th 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

$U_1$	$=$	$7$
$U_2$	$=$	$7+$ $7$	$=$	$14$
$U_3$	$=$	$14+$ $7$	$=$	$21$
$U_4$	$=$	$21+$ $7$	$=$	$28$

$7+14+21+28…$

Question 2 of 3

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

$(i)$ $d=$ (1) $\text(cm)$

$(ii)$ $S_n=$ (124.25) $\text(m)$

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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

$1 \text(metre)$

$=$

$100 \text(centimetres)$

$12425xx1/(100)$

$=$

$124.25 \text(cm)$

$(i) d=1 \text(cm)$

$(ii) S_n=124.25 \text(m)$

Question 3 of 3

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

$(i)$ $U_21=$ (41)

$(ii)$ $n=$ (31)

$(iii)$ $n=$ (36)

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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
$36$	$=$	$n$
$n$	$=$	$36$

$(i) U_21=41$

$(ii) n=31$

$(iii) n=36$

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General Rule of an Arithmetic Sequence

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Sum of an Arithmetic Sequence

General Rule of an Arithmetic Sequence

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Sum of an Arithmetic Sequence

General Rule of an Arithmetic Sequence

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