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

24 sample cards from the Functions, Sequences and Series deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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 $r$.

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

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

4. 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 $r \neq 1$.

5. 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 $S_{\infty} = \frac{a}{1-r}$.

6. 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).

7. 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 $n$.

8. 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 $n$).

9. 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.

10. 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.

11. 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.

12. 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.

13. 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 $u_{1}$.

14. 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.

15. 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 $L$.

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

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

17. 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 $n$.

18. 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.

19. 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: $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$.

20. 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.

21. 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 $40$.

22. 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 $(a+b)^{n}$.

23. 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).

24. 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.

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