Test of Mathematics for University Admission (TMUA) Functions, Sequences and Series Syllabus

Describe how to solve a 'hidden quadratic' exponential equation such as $e^{2x} - 5e^{x} + 6 = 0$.

Substitute $y = e^{x}$ to get $y^{2} - 5y + 6 = 0$, factorise to $(y-2)(y-3)=0$, so $y=2$ or $y=3$. Then $x = \ln 2$ or $x = \ln 3$.

How do you solve an equation where the unknown appears in exponents on both sides, e.g. $3^{x} = 5^{x-1}$?

Take logs of both sides: $x\ln 3 = (x-1)\ln 5$. Expand and collect $x$ terms: $x(\ln 3 - \ln 5) = -\ln 5$, so $x = \frac{-\ln 5}{\ln 3 - \ln 5} = \frac{\ln 5}{\ln 5 - \ln 3}$.

A quantity follows $y = ab^{x}$. How can logarithms linearise this for graphing?

Take logs: $\log y = \log a + x\log b$. Plotting $\log y$ against $x$ gives a straight line with gradient $\log b$ and intercept $\log a$.

A quantity follows a power law $y = ax^{n}$. How do you linearise it with logarithms?

Take logs: $\log y = \log a + n\log x$. Plotting $\log y$ against $\log x$ gives a straight line of gradient $n$ and intercept $\log a$.

In the model $\log y = mx + c$ obtained from $y = ab^{x}$, how do you recover $a$ and $b$?

Compare with $\log y = \log a + x\log b$: gradient $m = \log b$ so $b = 10^{m}$; intercept $c = \log a$ so $a = 10^{c}$ (using base-10 logs).

Define an arithmetic sequence and its common difference.

An arithmetic sequence is one in which consecutive terms differ by a constant, the common difference $d = a_{n+1} - a_{n}$. Each term is obtained by adding $d$ to the previous one.

State the formula for the $n$th term of an arithmetic sequence with first term $a$ and common difference $d$.

$u_{n} = a + (n-1)d$.

State the formula for the sum of the first $n$ terms of an arithmetic series.

$S_{n} = \frac{n}{2}\left[2a + (n-1)d\right]$, or equivalently $S_{n} = \frac{n}{2}(a + l)$ where $l$ is the last term.

Define a geometric sequence and its common ratio.

A geometric sequence is one in which each term is a constant multiple of the previous one; the common ratio $r = \frac{a_{n+1}}{a_{n}}$. Terms are found by multiplying by $r$.

State the formula for the $n$th term of a geometric sequence with first term $a$ and common ratio $r$.

$u_{n} = ar^{n-1}$.

State the formula for the sum of the first $n$ terms of a geometric series.

$S_{n} = \frac{a(1-r^{n})}{1-r}$ (or $\frac{a(r^{n}-1)}{r-1}$), valid for $r \neq 1$.

State the condition for a geometric series to converge and the formula for its sum to infinity.

It converges when $|r| < 1$. The sum to infinity is $S_{\infty} = \frac{a}{1-r}$.

Compare arithmetic and geometric sequences in how successive terms are generated.

Arithmetic: add a constant difference $d$ each step (linear growth). Geometric: multiply by a constant ratio $r$ each step (exponential growth/decay).

What does the sigma notation $\sum_{r=1}^{n} f(r)$ mean?

It denotes the sum $f(1) + f(2) + \cdots + f(n)$, adding $f(r)$ for each integer $r$ from the lower limit $1$ to the upper limit $n$.

Evaluate $\sum_{r=1}^{n} 1$ and explain.

$\sum_{r=1}^{n} 1 = n$, because the constant $1$ is added $n$ times (once for each value of $r$ from $1$ to $n$).

State the standard result for $\sum_{r=1}^{n} r$.

$\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$, the sum of the first $n$ positive integers.

How do you handle a constant multiple and a sum inside sigma notation, e.g. $\sum (ar + b)$?

Sigma is linear: $\sum_{r=1}^{n}(a f(r) + b\, g(r)) = a\sum_{r=1}^{n} f(r) + b\sum_{r=1}^{n} g(r)$. Constants factor out and sums split.

How do you evaluate a sum whose lower limit is not 1, such as $\sum_{r=4}^{10} f(r)$?

Use $\sum_{r=4}^{10} f(r) = \sum_{r=1}^{10} f(r) - \sum_{r=1}^{3} f(r)$, subtracting the unwanted lower terms.

What is a recurrence relation, and what extra information is needed to define a sequence with one?

A recurrence relation defines each term in terms of previous term(s), e.g. $u_{n+1} = f(u_{n})$. It requires one or more initial terms (e.g. $u_{1}$) to generate the sequence uniquely.

Write the arithmetic and geometric sequences as recurrence relations.

Arithmetic: $u_{n+1} = u_{n} + d$. Geometric: $u_{n+1} = r\,u_{n}$. Both with a given first term $u_{1}$.

Define a periodic sequence and how to identify its order from a recurrence.

A sequence is periodic if terms repeat in a fixed cycle: $u_{n+k} = u_{n}$ for all $n$. The period (order) $k$ is the length of the repeating block; find it by listing terms until they repeat.

What is the limit of a convergent recurrence $u_{n+1}=f(u_{n})$, and how do you find it?

If it converges to a limit $L$, then $L$ satisfies the fixed-point equation $L = f(L)$. Solve this equation for $L$.

State the Binomial Theorem for $(a+b)^{n}$ where $n$ is a positive integer.

$(a+b)^{n} = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r} = a^{n} + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^{2} + \cdots + b^{n}$.

Define the binomial coefficient $\binom{n}{r}$ in terms of factorials.

$\binom{n}{r} = {}^{n}C_{r} = \frac{n!}{r!\,(n-r)!}$, the number of ways to choose $r$ items from $n$.

What is the general term in the expansion of $(a+b)^{n}$?

The term in $b^{r}$ is $\binom{n}{r} a^{n-r} b^{r}$ (the $(r+1)$th term). Used to find a specific term without full expansion.

How does Pascal's triangle relate to binomial coefficients?

Row $n$ of Pascal's triangle gives the coefficients $\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}$. Each entry is the sum of the two above it: $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$.

State the symmetry property of binomial coefficients.

$\binom{n}{r} = \binom{n}{n-r}$, since choosing $r$ items to include is equivalent to choosing $n-r$ items to exclude.

Find the coefficient of $x^{3}$ in the expansion of $(2 + x)^{5}$.

Term: $\binom{5}{3} 2^{2} x^{3} = 10 \times 4 \times x^{3} = 40x^{3}$. The coefficient is $40$.

What is the sum of all binomial coefficients in row $n$, i.e. $\sum_{r=0}^{n}\binom{n}{r}$?

$\sum_{r=0}^{n}\binom{n}{r} = 2^{n}$, obtained by setting $a=b=1$ in $(a+b)^{n}$.

Application: A loan of £1000 grows by 5% interest per year. Express the amount after $n$ years and identify the type of sequence.

Amount $= 1000 \times 1.05^{n}$, a geometric sequence with first term (after year 1) $1050$ and common ratio $r = 1.05$ (compound growth).

Application: How is the binomial expansion used to approximate $(1.02)^{6}$ quickly?

Write as $(1+0.02)^{6} \approx 1 + 6(0.02) + \binom{6}{2}(0.02)^{2} = 1 + 0.12 + 15(0.0004) = 1.126$. Higher powers of the small term are negligible.