VertexMath · MCR3U · Free worksheet

Sequences and Series Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 50 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Test every consecutive pair: a constant difference $d=t_n-t_{n-1}$ means arithmetic; a constant ratio $r=\dfrac{t_n}{t_{n-1}}$ means geometric.
Arithmetic: $t_n=a+(n-1)d$ and $S_n=\dfrac{n}{2}(t_1+t_n)=\dfrac{n}{2}\left[2a+(n-1)d\right]$ .
Geometric: $t_n=ar^{n-1}$ and $S_n=\dfrac{a(r^n-1)}{r-1}$ , $r\ne1$ .
Pascal's triangle: row $n$ (the row $1, n, \dots, n, 1$ ) gives the coefficients of $(a+b)^n$ . Each entry is the sum of the two entries above it.
In $ar^{n-1}$ the exponent applies to $r$ only: $ar^{n-1}$ is not $(ar)^{n-1}$ . In the same way, $(3x)^2=9x^2$ , not $3x^2$ .

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Arithmetic · Geometric · Recursive · Arithmetic Series · Geometric Series

K1 Arithmetic Sequences

a)
The first four terms of an arithmetic sequence are graphed below. Each grid square is 1 unit. Determine $d$ and $t_{20}$ .
b)
Determine the general term of $41, 34, 27, \dots$ and find $t_{15}$ .
c)
Which term of $150, 143, 136, \dots$ is equal to $3$ ?

Stuck? Peek at a hint

Find $d$ as a later term minus the term before it. Keep $(n-1)d$ in brackets when $d$ is negative.

K2 Geometric Sequences

a)
Determine $r$ and $t_{10}$ for $3, -6, 12, \dots$
b)
A geometric sequence has $a=5$ and $r=3$ . Determine its general term and evaluate $t_6$ .
c)
Which term of $4, 12, 36, \dots$ is equal to $2916$ ?

Stuck? Peek at a hint

Divide a term by the one before it to find $r$ . In $ar^{n-1}$ , raise $r$ to the power first, then multiply by $a$ .

K3 Recursive Sequences

a)
Write $t_3$ to $t_6$ for $t_1=2$ , $t_2=5$ , $t_n=t_{n-1}+t_{n-2}$ .
b)
Determine the explicit formula for $t_1=9$ , $t_n=t_{n-1}+4$ .
c)
For the sequence in the table, state whether it is arithmetic, geometric or neither. Then write a recursive formula for it.

$n$ $1$ $2$ $3$ $4$ $5$
$t_n$ $3$ $7$ $15$ $31$ $63$

$n$	$1$	$2$	$3$	$4$	$5$
$t_n$	$3$	$7$	$15$	$31$	$63$

Stuck? Peek at a hint

Write out the first few terms before looking for a pattern. For (c), compute both the differences and the ratios of consecutive terms.

K4 Arithmetic Series

a)
Find the sum $7+11+15+\dots+87$ .
b)
Find $S_{25}$ for the arithmetic series with $a=-6$ and $d=3$ .
c)
An arithmetic series has $a=3$ and $d=4$ . How many terms add to $300$ ?

Stuck? Peek at a hint

Find $n$ from the general term before using the sum formula. Inside $2a+(n-1)d$ , multiply by $d$ before adding.

K5 Geometric Series

a)
Find $S_8$ for $2+6+18+\dots$
b)
How many terms of $5+10+20+\dots$ add to $1275$ ?
c)
Find $S_6$ for $64-32+16-\dots$

Stuck? Peek at a hint

Identify $a$ and $r$ first, including the sign of $r$ . In the sum formula, compute $r^n$ before anything else.

Real World (A)

Savings · Chain Messages · Bouncing Ball

A1 Laptop Fund

Noah saves $5.00 in the first week of the year. Each week after that, he saves $0.50 more than the week before.

a)
How much does Noah save in week 20?
b)
How much has he saved in total after 20 weeks?
c)
Noah wants $1000 for a laptop by the end of the year (52 weeks). Solve $S_n=1000$ . Can he reach his goal in time? How much does he have after 52 weeks?

Stuck? Peek at a hint

Write both amounts in dollars before choosing $a$ and $d$ .

A2 Chain Message

In round 1 of a chain message, 4 people receive it. In each later round, every person who received it in the previous round sends it to 3 people who have not seen it yet.

a)
Write the general term for the number of new people who receive the message in round $n$ . How many receive it in round 6?
b)
In which round do more than $100\,000$ new people first receive the message?
c)
Ontario has about 16 million people. In which round does the formula first give more new people than that? What does this mean for the model?

Stuck? Peek at a hint

Round 1 has $3^0$ in its formula, so check which exponent goes with which round.

A3 Bouncing Ball

A ball is dropped from a height of 10 m. After each bounce, it rises to 60% of the height it fell from.

a)
Find the height of the 4th rebound.
b)
Find the total vertical distance the ball has travelled when it hits the ground for the 6th time. Round to the nearest hundredth of a metre.
c)
The formula gives a positive rebound height for every bounce, so it predicts the ball never stops. Find the first rebound lower than 1 cm, and say what this means for the model.

Stuck? Peek at a hint

Sketch the ball's path for the first three bounces and label every distance it travels.

Think It Through (T)

Error Analysis · Building Sequences

T1 Spot the Flaw

A student expands $(3x-2)^4$ :
"Row 4 of Pascal's triangle is $1, 4, 6, 4, 1$ , so $(3x-2)^4=(3x)^4+4(3x)^3(-2)+6(3x)^2(-2)^2+4(3x)(-2)^3+(-2)^4$ , which is $3x^4-24x^3+72x^2-96x+16$ ."

a)
Evaluate $(3x-2)^4$ and the student's answer at $x=1$ . What does the result prove?
b)
Identify the error and write the correct expansion.
c)
Explain why the student's $x$ -term and constant term are correct anyway.

Stuck? Peek at a hint

For (b), compare $(3x)^2$ with $(3x)(3x)$ .

T2 Build It Backwards

Build each sequence from the information given. Some parts have more than one answer.

a)
An arithmetic sequence has $t_5=23$ and $t_{12}=51$ . Determine its general term.
b)
A geometric sequence has $t_2=6$ and $t_4=54$ . Determine every possible general term.
c)
Find every value of $x$ for which $2, x, 18$ is a geometric sequence, and every value for which it is an arithmetic sequence.

Stuck? Peek at a hint

Count the steps between the two given terms: from $t_5$ to $t_{12}$ , $d$ is added 7 times.

Boss Level

Optional challenge

Neither 3 nor 5

Use arithmetic series to answer each part.

a)
Find the sum of all multiples of 3 from 3 to 300.
b)
Find the sum of all multiples of 5 from 5 to 300.
c)
Find the sum of all integers from 1 to 300 that are divisible by neither 3 nor 5.

Stuck? Peek at a hint

For (c), start from $1+2+\dots+300$ , then decide which multiples to remove.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $d=3$ , $t_{20}=52$
b) $t_n=48-7n$ ; $t_{15}=-57$
c) $t_{22}$ (the 22nd term)

a) $r=-2$ , $t_{10}=-1536$
b) $t_n=5(3)^{n-1}$ ; $t_6=1215$
c) $t_7$ (the 7th term)

a) $7, 12, 19, 31$
b) $t_n=4n+5$
c) Neither; $t_1=3$ , $t_n=2t_{n-1}+1$

a) $987$ ( $21$ terms)
b) $750$
c) $12$ terms

a) $6560$
b) $8$ terms
c) $42$

Real World

a) $14.50
b) $195.00
c) $n^2+19n-4000=0$ gives $n\approx54.5$ (the negative root is not a number of weeks). He first passes $1000 in week 55, so no. After 52 weeks he has $923.00.

a) $t_n=4(3)^{n-1}$ ; $972$ people
b) Round 11 ( $236\,196$ people; round 10 has $78\,732$ )
c) Round 15 (about $19.1$ million). That is more new people than live in Ontario, so the model must break down before round 15: the message runs out of people who have not seen it.

a) $1.296$ m
b) $37.67$ m
c) The 14th rebound (about $0.78$ cm). Past this point the model no longer describes a real ball: a real ball stops bouncing after finitely many bounces.

Think It Through

a) $1$ and $-29$ . The two expressions disagree at $x=1$ , so the expansion is wrong.
b) $(3x)^k$ was written as $3x^k$ ; it is $3^kx^k$ . Correct: $81x^4-216x^3+216x^2-96x+16$
c) The $x$ -term contains $(3x)^1=3x$ , and the constant term contains no factor $3x$ , so writing $3x^k$ for $(3x)^k$ changes nothing there.

a) $t_n=4n+3$ ( $a=7$ , $d=4$ )
b) $t_n=2(3)^{n-1}$ or $t_n=-2(-3)^{n-1}$
c) Geometric: $x=6$ or $x=-6$ . Arithmetic: $x=10$ .

Boss Level

Boss

a) $15\,150$
b) $9150$
c) $24\,000$

Free cheat sheet

Sequences and Series Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

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Phone card

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Most testedFind the sum of the first $10$ terms of $3+6+12+\cdots$ . (card 6)

1Arithmetic sequences

$t_n=a+(n-1)d\qquad d=t_n-t_{n-1}$

UseCheck $d$ for every consecutive pair. Finding $n$ : solve for it; it must be a positive integer.

Trap $a=20$ , $d=-3$ : $t_{15}=20+14(-3)=-22$ , not $20+14-3$ .

Trap$45 000 now, $+$ $1500 a year: after $10$ years is $45\,000+10(1500)$ , the $11$ th term.

2Geometric sequences

$t_n=ar^{n-1}\qquad r=\dfrac{t_n}{t_{n-1}}$

UseCheck $r$ for every pair. The exponent applies to $r$ only. Growth of $4\%$ : $r=1.04$ .

Trap $a=5$ , $r=2$ : $t_4=5(2)^3=40$ , not $(5\cdot2)^3=1000$ .

Trap $486=6(3)^{n-1}$ : $3^{n-1}=81=3^4$ , so $n-1=4$ and $n=5$ .

3Recursive formulas

$t_1=a,\ \ t_n=t_{n-1}+d\qquad t_1=a,\ \ t_n=r\,t_{n-1}$

UseTo go explicit, keep the same $a$ and $d$ (or $r$ ) and keep $n-1$ .

Example $t_1=15$ , $t_n=t_{n-1}-6$ : $t_n=15-6(n-1)=21-6n$ .

Trap $t_1=1$ , $t_2=3$ , $t_n=t_{n-1}+t_{n-2}$ gives $1,3,4,7,11$ : start from the given terms, not Fibonacci's.

4Classifying

UseTest differences and ratios. Constant difference: arithmetic. Constant ratio: geometric. Neither: look for another rule.

Trap $4, 8, 16, 32$ : the differences change, but the ratio is $2$ , so it is geometric, not "neither".

5Arithmetic series

$S_n=\dfrac n2(t_1+t_n)=\dfrac n2\left[2a+(n-1)d\right]$

UseFind $n$ first. Solving for $n$ can give a quadratic: keep only positive integers. Use one unit (dollars or cents).

Example $5+8+\cdots+62$ : $62=5+3(n-1)$ , $n=20$ , $S_{20}=10(67)=670$ .

TrapIn $2a+(n-1)d$ , multiply $(n-1)d$ first, then add $2a$ .

6Geometric series

$S_n=\dfrac{a(r^n-1)}{r-1},\quad r\ne1$

UseBouncing ball: the first drop, then each bounce counts up and down.

Example $3+6+12+\cdots$ : $S_{10}=3(2^{10}-1)/(2-1)=3069$ .

Trap $3^n=243$ gives $n=5$ , not $243/3=81$ : find the power.

TrapA ball dropped $8$ m, rebounding to $3/4$ : by the 4th landing it travels $8+2(6+4.5+3.375)=35.75$ m.

7Pascal's triangle

UseRow $n$ ( $1, n, \dots, n, 1$ ) gives the coefficients of $(a+b)^n$ . Powers of $a$ fall as powers of $b$ rise; each term's exponents add to $n$ .

Example $(x+2)^3=x^3+3x^2(2)+3x(2)^2+2^3=x^3+6x^2+12x+8$ .

TrapIn $(2x-3)^3$ , $(2x)^3=8x^3$ , not $2x^3$ : the power applies to the coefficient too.

Which sequence?

Constant difference Arithmetic

Constant ratio Geometric

Each term from the two before Recursive

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