MCR3U · Unit 3 test review

Discrete Functions: Test Review

Everything for the Unit 3 test on one page: the key facts from all 8 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Sequences as Discrete Functions

A sequence is a discrete function: its domain is ${1, 2, 3, \dots}$ , and $t_{n}$ denotes the value of the $n$ th term.
An explicit formula gives $t_{n}$ directly in terms of $n$ — no previous terms required.
A recursive formula defines $t_{n}$ using earlier term(s), and must state a starting value such as $t_{1}$ .
Constant first differences $\Rightarrow$ arithmetic; constant ratio $\Rightarrow$ geometric; constant second differences $\Rightarrow$ quadratic-type.
Arithmetic recursion $t_{n} = t_{n - 1} + d$ matches $t_{n} = a + (n - 1) d$ ; geometric recursion $t_{n} = r t_{n - 1}$ matches $t_{n} = a r^{n - 1}$ .
The graph of a sequence is a set of separate points $(n, t_{n})$ , never a connected curve.

Arithmetic Sequences

Common difference: $d = t_{n} - t_{n - 1}$ (same value between every pair of consecutive terms).
Explicit formula: $t_{n} = a + (n - 1) d$ , where $a = t_{1}$ .
If $d > 0$ the sequence is increasing; if $d < 0$ it is decreasing; if $d = 0$ every term is identical.
Given two non-consecutive terms $t_{i}$ and $t_{j}$ : $d = \frac{t _{j} - t _{i}}{j - i}$ .
$t_{n} = d n + (a - d)$ is a linear function of $n$ with slope $d$ — arithmetic sequences are sampled lines.
Number of terms from $a$ to a last term $L$ : $n = \frac{L - a}{d} + 1$ .

Geometric Sequences

Common ratio: $r = \frac{t _{n}}{t _{n - 1}}$ (same value between every pair of consecutive terms).
Explicit formula: $t_{n} = a r^{n - 1}$ , where $a = t_{1}$ .
$∣ r ∣ > 1$ gives growth; $0 < ∣ r ∣ < 1$ gives decay toward 0; $r < 0$ makes the sequence alternate sign.
Given two non-consecutive terms $t_{i}$ and $t_{j}$ ( $j > i$ ): $r^{j - i} = \frac{t _{j}}{t _{i}}$ , so take the $(j - i)$ th root; if $j - i$ is even, both $\pm$ roots are valid ratios.
A $k %$ increase gives $r = 1 + \frac{k}{100}$ ; a $k %$ decrease gives $r = 1 - \frac{k}{100}$ .
To find $n$ , isolate $r^{n - 1}$ and match bases: $3 (2)^{n - 1} = 1536 \Rightarrow 2^{n - 1} = 2^{9} \Rightarrow n = 10 .$

Arithmetic Series

A series is the sum of a sequence's terms: $S_{n} = t_{1} + t_{2} + \dots + t_{n}$ .
Sum formula (needs last term): $S_{n} = \frac{n}{2} (a + t_{n})$ .
Sum formula (needs only $a$ and $d$ ): $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ .
Gauss's trick: pairing the first and last terms of an arithmetic series always gives the same sum, $a + t_{n}$ , and there are $\frac{n}{2}$ pairs.
Count terms before summing: solve $a + (n - 1) d = t_{n}$ for $n$ (the number of gaps plus 1).
A single term from partial sums: $t_{n} = S_{n} - S_{n - 1}$ ; and $1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$ .

Geometric Series

Geometric series sum: $S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}$ (equivalently $\frac{a ( 1 - r ^{n} )}{1 - r}$ ) for $r \neq = 1$ .
Special case: if $r = 1$ , every term equals $a$ , so $S_{n} = na$ .
Derivation idea: $S_{n} - r S_{n}$ cancels every term except the first and the shifted last term.
$a$ is the first term, $r$ is the common ratio, $n$ is how many terms are being added.
If the last term is given, find $n$ from $a r^{n - 1} = t_{n}$ first, or use $S_{n} = \frac{r t _{n} - a}{r - 1}$ .

Pascal's Triangle and the Binomial Theorem

Each entry in Pascal's triangle is the sum of the two entries diagonally above it; every row starts and ends with 1.
Row $n$ sums to $2^{n}$ .
Row $n$ gives the coefficients for expanding $(a + b)^{n}$ .
In each term of the expansion, the exponent of $a$ decreases from $n$ to $0$ while the exponent of $b$ increases from $0$ to $n$ — always summing to $n$ .
Recursive rule with rows and positions counted from 0: $t_{n, r} = t_{n - 1, r - 1} + t_{n - 1, r} .$
The term of $(a + b)^{n}$ containing $b^{r}$ is $t_{n, r} a^{n - r} b^{r}$ ; there are $n + 1$ terms in all.

Simple and Compound Interest

Simple interest: $I = P r t$ , and the final amount is $A = P (1 + r t)$ — grows arithmetically (linearly).
Compound interest: $A = P (1 + i)^{n}$ — grows geometrically, with common ratio $(1 + i)$ .
$i = \frac{r}{m}$ (rate per period) and $n = m t$ (number of periods), where $m$ = compounding periods per year.
Present value (solving backward): $P = A (1 + i)^{- n}$ .
Simple interest balances form an arithmetic sequence ( $d = P r$ ); compound balances form a geometric sequence ( $r = 1 + i$ ).
In $I = P r t$ , time must be in years: 9 months is $t = 0.75$ . Round money to the cent only at the end.

Simple Annuities

Ordinary simple annuity: equal payments $R$ at the end of each period, with the payment interval matching the compounding interval.
Future value: $F V = R \cdot \frac{( 1 + i ) ^{n} - 1}{i}$ .
Present value: $P V = R \cdot \frac{1 - ( 1 + i ) ^{- n}}{i}$ .
$n$ is the TOTAL number of payments, and $i$ is the interest rate per payment period — same matching rule as compound interest.
Both formulas come from a geometric series: the future value adds $R + R (1 + i) + \dots + R (1 + i)^{n - 1}$ .
Total paid $= n R$ . Loan interest $= n R - P V$ ; savings interest $= F V - n R$ .

Where marks get lost

Treating $n$ like a continuous variable (plugging in $n = 2.5$ ) — $n$ must be a positive integer. (Sequences as Discrete Functions)
Writing a recursive rule with no starting term — the recursion $t_{n} = t_{n - 1} + 3$ alone is meaningless without knowing $t_{1}$ . (Sequences as Discrete Functions)
Committing to a formula after seeing only two terms, instead of verifying against several. (Sequences as Discrete Functions)
Mixing up $n$ (the position) with $t_{n}$ (the value) — e.g. saying 'the 4th term is 4' when the question is asking for $t_{4}$ , which could be anything. (Sequences as Discrete Functions)
Using the wrong $n$ inside a recursive rule that contains $n$ : for $t_{1} = 1, t_{n} = t_{n - 1} + 2 n$ , the third term is $t_{3} = t_{2} + 2 (3)$ , using the current position 3, not $t_{2} + 2 (2)$ . (Sequences as Discrete Functions)
Adding $d$ exactly $n$ times instead of $(n - 1)$ times — you don't add anything to reach $t_{1}$ itself. (Arithmetic Sequences)
Computing $d$ from two terms that aren't consecutive without dividing by the gap in position (e.g. treating $t_{9} - t_{4}$ as $d$ instead of $\frac{t _{9} - t _{4}}{5}$ ). (Arithmetic Sequences)
Forgetting that $d$ can be negative — a decreasing sequence is still arithmetic. (Arithmetic Sequences)
Rounding $n$ to the nearest integer when it doesn't come out whole, instead of concluding the given value isn't actually a term of the sequence. (Arithmetic Sequences)
Counting terms by subtracting and dividing but forgetting the final $+ 1$ : from 7 to 203 in steps of 4 there are $196 \div 4 = 49$ gaps, which means 50 terms, not 49. (Arithmetic Sequences)
Multiplying by $r$ exactly $n$ times instead of $(n - 1)$ times. (Geometric Sequences)
Forgetting that a decay ratio less than 1 is still called $r$ , not treated like a 'negative difference' — there's no subtraction happening at all. (Geometric Sequences)
For a percentage decrease of $k %$ , using $r = k$ instead of $r = 1 - \frac{k}{100}$ (and for a $k %$ increase, $r = 1 + \frac{k}{100}$ , not $r = k$ ). (Geometric Sequences)
When finding $r$ from two non-consecutive terms, forgetting to take the correct root (using the gap in position, not just dividing the two values). (Geometric Sequences)
Keeping only $r = 3$ when $r^{2} = 9$ : an even gap between the given terms allows $r = - 3$ as well, giving a second, alternating sequence. (Geometric Sequences)
Confusing the sequence formula $t_{n} = a + (n - 1) d$ with the series formula $S_{n}$ — a series is a SUM, a sequence term is a single VALUE. (Arithmetic Series)
Using $n$ instead of $(n - 1)$ inside the $2 a + (n - 1) d$ version of the formula. (Arithmetic Series)
Forgetting to divide by 2 (or dividing by $n$ instead of 2) when applying $S_{n} = \frac{n}{2} (a + t_{n})$ . (Arithmetic Series)
Trying to use the arithmetic series formula on a sequence that isn't actually arithmetic (no constant $d$ ) — always confirm the common difference first. (Arithmetic Series)
Counting terms as $(t_{n} - a) \div d$ : for $4 + 9 + \dots + 99$ that gives 19, which is the number of gaps. There are $19 + 1 = 20$ terms. (Arithmetic Series)
Plugging $r = 1$ into $\frac{a ( r ^{n} - 1 )}{r - 1}$ and dividing by zero, instead of using $S_{n} = na$ for that special case. (Geometric Series)
Mixing up $n$ (number of terms being summed) with the exponent used in the LAST term's formula, $t_{n} = a r^{n - 1}$ — the series formula's exponent on $r$ is $n$ , not $n - 1$ . (Geometric Series)
Sign errors when both numerator and denominator are negative (e.g. $r < 1$ using the $r^{n} - 1$ form) — switch to the $1 - r^{n}$ form to avoid double negatives. (Geometric Series)
Forgetting that the geometric series formula requires a genuinely constant ratio — check $r$ is the same throughout before applying it. (Geometric Series)
Substituting the last term's value for $n$ : in $3 + 6 + \dots + 3072$ , $n$ is 11 (found from $3 (2)^{n - 1} = 3072$ ), not 3072. For bouncing balls, also forgetting to double the rebound heights. (Geometric Series)
Using the wrong row of Pascal's triangle — row $n$ (not row $n + 1$ or $n - 1$ ) belongs to $(a + b)^{n}$ , and row 0 is the very top '1'. (Pascal's Triangle and the Binomial Theorem)
Forgetting that the exponents of $a$ and $b$ must add to $n$ in every single term. (Pascal's Triangle and the Binomial Theorem)
Dropping the alternating signs when expanding $(a - b)^{n}$ — the sign flips on every odd-powered term of $b$ . (Pascal's Triangle and the Binomial Theorem)
Forgetting to raise the COEFFICIENT of $a$ or $b$ to the power too, e.g. treating $(2 x)^{3}$ as $2 x^{3}$ instead of $8 x^{3}$ . (Pascal's Triangle and the Binomial Theorem)
Giving only the Pascal entry as a coefficient: in $(2 x + 3)^{5}$ the coefficient of $x^{3}$ is $10 (2^{3}) (3^{2}) = 720$ , not 10. (Pascal's Triangle and the Binomial Theorem)
Using the annual rate $r$ directly as $i$ when compounding happens more than once a year — always divide by $m$ first. (Simple and Compound Interest)
Using the number of YEARS as $n$ when compounding is not annual — always multiply by $m$ to get the number of periods. (Simple and Compound Interest)
Confusing simple interest ( $I = P r t$ , straight-line growth) with compound interest ( $A = P (1 + i)^{n}$ , geometric growth) — simple interest problems never use an exponent. (Simple and Compound Interest)
Forgetting that $A$ in $I = P r t$ questions is $P + I$ , not just $I$ — read carefully whether the question asks for the interest earned or the total final amount. (Simple and Compound Interest)
Rounding $i$ too early (e.g. rounding $0.05/12$ to $0.004$ ) — keep several extra decimal places until the final answer to avoid compounding a rounding error. (Simple and Compound Interest)
Confusing future value (how much accumulates) with present value (equivalent lump sum today) — decide which one the question is actually asking for before picking a formula. (Simple Annuities)
Using the annuity formula's $(1 + i)^{n} - 1$ but forgetting to divide by $i$ at the end. (Simple Annuities)
Mismatching the payment period and the compounding period — an ordinary SIMPLE annuity requires them to be the same (monthly payments need monthly compounding). (Simple Annuities)
Applying the compound interest formula $A = P (1 + i)^{n}$ to a series of payments — that formula is for a single lump sum, not repeated deposits. (Simple Annuities)
Rounding $i$ too early in a long calculation, which can shift the final dollar amount noticeably over many periods. (Simple Annuities)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Sequences as Discrete Functions
1
For $t_{n} = 2 n + 1$ , what is $t_{5}$ ?
Sequences as Discrete Functions
2
Match each description to the correct term.
1. A. $n$
2. B. $t_{n}$
3. C.Explicit formula
4. D.Recursive formula
Position number in the list
Value stored at that position
Formula using only $n$
Formula using $t_{n - 1}$
Sequences as Discrete Functions
3
The Fibonacci sequence is $t_{1} = t_{2} = 1, t_{n} = t_{n - 1} + t_{n - 2} .$ What is $t_{7}$ ?
Arithmetic Sequences
4
An arithmetic sequence has $a = 7$ and $d = 4$ . Find $t_{12}$ .
Arithmetic Sequences
5
Using $a = 5, d = 5$ (from the worked example), find $t_{20}$ .
Arithmetic Sequences
6
Match each sequence to its common difference.
1. A. $- 4$
2. B. $0.5$
3. C. $3$
4. D. $5$
$5, 8, 11, \dots$
$20, 16, 12, \dots$
$1.5, 2, 2.5, \dots$
$- 7, - 2, 3, \dots$
Geometric Sequences
7
A geometric sequence has $a = 5, r = 3$ . Find $t_{6}$ .
Geometric Sequences
8
Using $r = 2$ and $t_{3} = 20$ from the previous question, find $a$ .
Geometric Sequences
9
Match each sequence to its common ratio.
1. A. $- 2$
2. B. $\frac{1}{2}$
3. C. $1.05$
4. D. $3$
$2, 6, 18, \dots$
$80, 40, 20, \dots$
$5, - 10, 20, \dots$
$100, 105, 110.25, \dots$
Arithmetic Series
10
Find $1 + 2 + 3 + \dots + 50$ using Gauss's method.
Arithmetic Series
11
Find the sum of the first 40 positive even numbers ( $2 + 4 + 6 + \dots + 80$ ).
Arithmetic Series
12
Match each series to its sum.
$1 + 2 + \dots + 10$
$2 + 4 + \dots + 20$
$1 + 3 + 5 + \dots + 19$
$10 + 20 + \dots + 100$
Geometric Series
13
Find the sum of the first 6 terms of a geometric sequence with $a = 1, r = 2$ .
Geometric Series
14
Every term of a sequence is $12$ (constant, so $r = 1$ ). Find $S_{9}$ .
Geometric Series
15
Match each geometric series to its sum.
$1 + 3 + 9 + 27$
$4 - 8 + 16 - 32$
$81 + 27 + 9 + 3 + 1$
$5 + 5 + 5 + 5 + 5 + 5$
Pascal's Triangle and the Binomial Theorem
16
What is the sum of the entries in row 5 of Pascal's triangle?
Pascal's Triangle and the Binomial Theorem
17
Expand $(x + 2)^{4}$ (from the worked example) and evaluate it at $x = 1$ to check. What value do you get?
Pascal's Triangle and the Binomial Theorem
18
Match each power to its expansion.
1. A. $4 x^{2} + 4 x + 1$
2. B. $x^{2} + 2 x + 1$
3. C. $x^{3} - 3 x^{2} + 3 x - 1$
4. D. $x^{3} + 3 x^{2} + 3 x + 1$
$(x + 1)^{2}$
$(x + 1)^{3}$
$(x - 1)^{3}$
$(2 x + 1)^{2}$
Simple and Compound Interest
19
Find the simple interest on 2500 dollars at 4%/year for 5 years.
dollars
Simple and Compound Interest
20
How much must be invested today at 5%/year compounded annually to grow to 3000 dollars in 4 years (round to the nearest cent)?
dollars
Simple and Compound Interest
21
For 8%/year over 3 years, match each compounding frequency to its values of i and n.
1. A. $i = \frac{0.08}{12}, n = 36$
2. B. $i = 0.02, n = 12$
3. C. $i = 0.04, n = 6$
4. D. $i = 0.08, n = 3$
Annually
Semi-annually
Quarterly
Monthly
Simple Annuities
22
100 dollars is deposited at the end of every month for 1 year into an account earning 6%/year compounded monthly. Find the future value (round to the nearest cent).
dollars
Simple Annuities
23
150 dollars is deposited at the end of every month for 2 years at 4%/year compounded monthly. Find the future value (round to the nearest cent).
dollars
Simple Annuities
24
Match each question to the quantity you should calculate.
1. A. $R$ from the $F V$ formula
2. B. $R$ from the $P V$ formula
3. C.Future value $F V$
4. D.Present value $P V$
How much will my monthly deposits grow to?
How big a loan can these payments repay?
What deposit reaches my savings goal?
What payment pays off this loan?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…

Discrete Functions: Test Review

Key facts

Where marks get lost

Practice test

Endless drill

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