Mathematics Admissions Test (MAT) Exponentials, Logarithms and Sequences Flashcards

24 sample cards from the Exponentials, Logarithms and Sequences deck

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

1. Under what condition does an infinite geometric series converge, and what is its sum?

   It converges when $|r| < 1$, and the sum to infinity is $S_{\infty} = \dfrac{a}{1 - r}$.

2. Find the sum to infinity of the geometric series $8 + 4 + 2 + 1 + \cdots$.

   Here $a=8$, $r=\frac{1}{2}$, so $S_{\infty} = \dfrac{8}{1 - \frac{1}{2}} = 16$.

3. In a geometric progression, how is each term related to its neighbours (geometric mean property)?

   The square of any term equals the product of its neighbours: $a_{n}^{2} = a_{n-1}\,a_{n+1}$, i.e. $a_{n} = \sqrt{a_{n-1}a_{n+1}}$.

4. In an arithmetic progression, how is each term related to its neighbours (arithmetic mean property)?

   Each term is the average of its neighbours: $a_{n} = \dfrac{a_{n-1} + a_{n+1}}{2}$.

5. What is a recurrence relation (recursively defined sequence)?

   A rule that defines each term using one or more previous terms together with initial value(s), e.g. $a_{n+1} = f(a_{n})$ with $a_{1}$ given.

6. Express an arithmetic sequence as a recurrence relation.

   $a_{n+1} = a_{n} + d$, with $a_{1} = a$ given.

7. Express a geometric sequence as a recurrence relation.

   $a_{n+1} = r\,a_{n}$, with $a_{1} = a$ given.

8. What is a fixed point (limit) of an iteration $x_{n+1} = f(x_{n})$, and how is it found?

   A fixed point $L$ satisfies $L = f(L)$; if the iteration converges, its limit is found by solving this equation.

9. Define a periodic sequence and give the meaning of its order.

   A sequence that repeats: $a_{n+k} = a_{n}$ for all $n$. The smallest such $k$ is the order (period) of the sequence.

10. What does the recurrence $x_{n+1} = \tfrac{1}{2}\!\left(x_{n} + \tfrac{N}{x_{n}}\right)$ converge to?

    It converges to $\sqrt{N}$ (the Newton-Raphson / Babylonian iteration for square roots).

11. What does sigma notation $\sum_{r=1}^{n} a_{r}$ mean?

    It denotes the sum $a_{1} + a_{2} + \cdots + a_{n}$; $r$ is the index, $1$ the lower limit and $n$ the upper limit.

12. Evaluate $\sum_{r=1}^{4} (2r + 1)$.

    $(3) + (5) + (7) + (9) = 24$.

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

    $\sum_{r=1}^{n} r = \dfrac{n(n+1)}{2}$.

14. State the standard summation result for $\sum_{r=1}^{n} r^{2}$.

    $\sum_{r=1}^{n} r^{2} = \dfrac{n(n+1)(2n+1)}{6}$.

15. State the standard summation result for $\sum_{r=1}^{n} r^{3}$.

    $\sum_{r=1}^{n} r^{3} = \dfrac{n^{2}(n+1)^{2}}{4} = \left(\dfrac{n(n+1)}{2}\right)^{2}$.

16. What is $\sum_{r=1}^{n} c$ where $c$ is a constant?

    $\sum_{r=1}^{n} c = nc$ (the constant added $n$ times).

17. State the linearity property of sigma notation.

    $\sum_{r=1}^{n} (\alpha a_{r} + \beta b_{r}) = \alpha\sum_{r=1}^{n} a_{r} + \beta\sum_{r=1}^{n} b_{r}$ for constants $\alpha,\beta$.

18. How can a sum starting from $r=k$ be expressed using sums starting from $r=1$?

    $\sum_{r=k}^{n} a_{r} = \sum_{r=1}^{n} a_{r} - \sum_{r=1}^{k-1} a_{r}$.

19. Use standard results to find $\sum_{r=1}^{n} (r^{2} + r)$ in factored form.

    $\sum_{r=1}^{n} r^{2} + \sum_{r=1}^{n} r = \dfrac{n(n+1)(2n+1)}{6} + \dfrac{n(n+1)}{2} = \dfrac{n(n+1)(n+2)}{3}$.

20. What is the noteworthy relationship between $\sum_{r=1}^{n} r^{3}$ and $\sum_{r=1}^{n} r$?

    $\sum_{r=1}^{n} r^{3} = \left(\sum_{r=1}^{n} r\right)^{2}$; the sum of cubes equals the square of the sum of the first $n$ integers.

21. Compare the long-term behaviour of arithmetic versus geometric growth.

    Arithmetic grows linearly (constant added each step), so it is unbounded but slow; geometric with $|r|>1$ grows exponentially and eventually dominates any arithmetic sequence.

22. How do you determine the number of terms $n$ in an arithmetic sequence given the last term $l$?

    Use $l = a + (n-1)d$ and solve for $n$: $n = \dfrac{l - a}{d} + 1$.

23. How can a recurring decimal such as $0.\overline{27}$ be evaluated as an infinite geometric series?

    $0.\overline{27} = \dfrac{27}{100} + \dfrac{27}{100^{2}} + \cdots = \dfrac{27/100}{1 - 1/100} = \dfrac{27}{99} = \dfrac{3}{11}$.

24. In compound-growth problems, how is the value after $n$ periods at rate $i$ modelled, and which type of sequence is it?

    $A_{n} = A_{0}(1+i)^{n}$, a geometric sequence with first term $A_{0}$ and common ratio $r = 1+i$.

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Exponentials, Logarithms and Sequences flashcards FAQ