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Chapter 1: Problem 16

Write out the first 12 or so powers of 4 : $$ 4,16,64,256,1024,4096,16384,65536, \ldots $$ Now create two sequences: the sequence of final digits: \(4,6,4,6,4,6, \ldots\) the sequence of leading digits: \(4,1,6,2,1,4,1,6, \ldots\) Both sequences seem to consist of a single "block", which repeats over and over for ever. (a) How long is the apparent repeating block for the first sequence? How long is the apparent repeating block for the second sequence? (b) It may not be immediately clear whether either of these sequences really repeats forever. Nor may it be clear whether the two sequences are alike, or whether one is quite different from the other. Can you give a simple proof that one of these sequences recurs, that is, repeats forever? (c) Can you explain why the other sequence seems to recur, and decide whether it really does recur forever?

Short Answer

Expert verified
The final digit sequence repeats every 2 terms, and this repetition is guaranteed as digit multiplication follows a predictable cycle. The leading digit sequence appears to repeat every 4 terms, but real consistent infinite repeat relies on cyclical behavior characteristic of base digit aggregations.

Step by step solution

01

Identify the final digit sequence

The final digit of each power of 4 forms the sequence: 4, 6, 4, 6, 4, 6, ... It becomes clear that the final digits repeat every 2 terms.
02

Identify the leading digit sequence

The leading digit of each power of 4 forms the initial sequence: 4, 1, 6, 2, 1, 4, 1, 6, ... Analyzing this sequence, it appears to repeat every 4 terms.
03

Prove the repetition of final digit sequence

Powers of 4 alternate between even and odd multiples of 4, affecting only the final digit. Namely, from 4 (even since 4^1=4) to 64 (even since 4^3=64) and across the cycle. This confirms the recurring pattern is achievable through alternating cycles of even number manipulation.
04

Explore reason for leading digit sequence pattern

For leading digits of powers of any number, the sequence seems irregular because the leading digit results from multiplying a number repeatedly by 4 while adjusting for scales in powers of 10. Although this initially appears repetitive, rigor reveals alignment with a more extended stage due to varying digit widths with higher powers bothering infinite clarity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Powers of Four
Understanding powers of four involves repeatedly multiplying the number 4 by itself. In mathematical terms, these are expressed as
  • \(4^1 = 4\),
  • \(4^2 = 16\),
  • \(4^3 = 64\),
  • \(4^4 = 256\), and so forth.
Each subsequent power grows rapidly. This swift expansion means the numbers are often much larger when you get to higher powers. It's essential to keep in mind how quickly these numbers increase, making their behavior, such as leading and final digits, quite interesting to examine.
This is because powers of four have significant uses in various fields, like computer science and geometry, and knowing how they behave can provide insightful patterns.
Leading Digits
Leading digits refer to the first digit of a number, and they can reveal interesting patterns when it comes to powers of four. Consider how we typically proceed by multiplying the number by four, and this often shifts the magnitude (number of digits) of the product.
For instance, going through the powers of four, you notice the sequence:
  • 4,
  • 1,
  • 6,
  • 2,
  • 1,
  • 4,
  • 1,
  • 6,
This sequence highlights a repeating block every four terms. This repetition unveils a crucial and often surprising regularity caused by the scaling effect when continuously multiplying by four. Each shift influences the numerically significant leading digit, especially as numbers pass the thresholds that increase the number of digits.
Final Digits
The sequence of final digits is determined by observing the last digit in each power of four. You would notice that it alternates between 4 and 6 as seen here:
  • 4,
  • 6,
  • 4,
  • 6,
This pattern repeats every two terms, showing a stable and predictable cycle. The reason for this alternation lies in the multiplication properties within the base 10 system.
Since 4 multiplied by an odd number results in a last digit of 4, and then by an even number results in a last digit of 6, the cycle perpetuates. Recognizing this reveals an important mathematical property that can deepen understanding of how powers function within different numeric bases.
Recurring Patterns
Recurring patterns are a significant aspect when analyzing sequences, especially within the context of number powers. When looking at the powers of four, both the leading and final digits present repeating patterns.
The recurring sequence of the leading digits - 4, 1, 6, 2 - emphasizes a four-term cycle. Meanwhile, the final digit pattern repeats with every two terms, oscillating predictably between 4 and 6. These patterns arise directly from the way multiplication by four reintegrates the numbers into round positional shifts within the numeric base system.
Understanding these recurring patterns helps in predicting future terms in the sequences and offers a glance into the mathematical backbone guiding these numerals. Such repeats are not only intriguing but also crucial for practical applications in number theory and computational mathematics.

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Most popular questions from this chapter

(a) The operation of "squaring" is a function: it takes a single real number \(x\) as input, and delivers a definite real number \(x^{2}\) as output. \- Every positive number arises as an output ("is the square of something" ). \(-\) Since \(x^{2}=(-x)^{2},\) each output (other than 0 ) arises from at least two different inputs. \- If \(a^{2}=b^{2},\) then \(0=a^{2}-b^{2}=(a-b)(a+b)\), so either \(a=b\), or \(a=-b\). Hence no two positive inputs have the same square, so each output (other than 0 ) arises from exactly two inputs (one positive and one negative). \- Hence each positive output \(y\) corresponds to just one positive input, called \(\sqrt{y}\). Find: (i) \(\sqrt{49}\) (ii) \(\sqrt{144}\) (iii) \(\sqrt{441}\) (iv) \(\sqrt{169}\) (v) \(\sqrt{196}\) (vi) \(\sqrt{961}\) (vii) \(\sqrt{96100}\) (b) Let \(a>0\) and \(b>0\). Then \(\sqrt{a b}>0\), and \(\sqrt{a} \times \sqrt{b}>0\), so both expressions are positive. Moreover, they have the same square, since $$ (\sqrt{a b})^{2}=a b=(\sqrt{a})^{2} \cdot(\sqrt{b})^{2}=(\sqrt{a} \times \sqrt{b})^{2} $$ \(\therefore \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}\) Use this fact to simplify the following: (i) \(\sqrt{8}\) (ii) \(\sqrt{12}\) (iii) \(\sqrt{50}\) (iv) \(\sqrt{147}\) (v) \(\sqrt{288}\) (vi) \(\sqrt{882}\) (c) [This part requires some written calculation.] Exact expressions involving square roots occur in many parts of elementary mathematics. We focus here on just one example - namely the regular pentagon. Suppose that a regular pentagon \(A B C D E\) has sides of length \(1 .\) (i) Prove that the diagonal \(A C\) is parallel to the side \(E D\). (ii) If \(A C\) and \(B D\) meet at \(X,\) explain why \(A X D E\) is a rhombus. (iii) Prove that triangles \(A D X\) and \(C B X\) are similar. (iv) If \(A C\) has length \(x\), set up an equation and find the exact value of \(x\). (v) Find the exact length of \(B X\). (vi) Prove that triangles \(A B D\) and \(B X A\) are similar. (vii) Find the exact values of \(\cos 36^{\circ}, \cos 72^{\circ}\). (viii) Find the exact values of \(\sin 36^{\circ}, \sin 72^{\circ}\).

The three integers \(a=3, b=4, c=5\) in the Pythagorean triple (3,4,5) form an arithmetic progression: that is, \(c-b=b-a\). Find all Pythagorean triples \((a, b, c)\) which form an arithmetic progression \(-\) that is, for which \(c-b=b-a\)

(a) Expand and simplify in your head: (i) \((\sqrt{2}+1)^{2}\) (ii) \((\sqrt{2}-1)^{2}\) (iii) \((1+\sqrt{2})^{3}\) (b) Simplify: (i) \(\sqrt{10+4 \sqrt{6}}\) (ii) \(\sqrt{5+2 \sqrt{6}}\) (iii) \(\sqrt{\frac{3+\sqrt{5}}{2}}\) (iv) \(\sqrt{10-2 \sqrt{5}}\) The expressions which occur in exercises to develop fluency in working with surds often appear arbitrary. But they may not be. The arithmetic of surds arises naturally: for example, some of the expressions in the previous problem have already featured in Problem \(\mathbf{3}(\mathrm{c}) .\) In particular, surds will feature whenever Pythagoras' Theorem is used to calculate lengths in geometry, or when a proportion arising from similar triangles requires us to solve a quadratic equation. So surd arithmetic is important. For example: \- A regular octagon with side length 1 can be surrounded by a square of side \(\sqrt{2}+1\) (which is also the diameter of its incircle); so the area of the regular octagon equals \((\sqrt{2}+1)^{2}-1\) (the square minus the four corners). \- \(\sqrt{2}-1\) features repeatedly in the attempt to apply the Euclidean algorithm, or anthyphairesis, to express \(\sqrt{2}\) as a "continued fraction". -\(\sqrt{10-2 \sqrt{5}}\) may look like an arbitrary, uninteresting repeated surd, but is in fact very interesting, and has already featured as \(4 \sin 36^{\circ}\) in Problem \(\mathbf{3}(\mathrm{c})\) \- One of the simplest ruler and compasses constructions for a regular pentagon \(A B C D E\) (see Problem 185) starts with a circle of radius 2 , centre \(O,\) and a point \(A\) on the circle, and in three steps constructs the next point \(B\) on the circle, where \(\underline{A B}\) is an edge of the inscribed regular pentagon, and $$ \underline{A B}=\sqrt{10-2 \sqrt{5}} $$

(a) In a sale which offers "15\% discount on all marked prices" I buy three articles: a pair of trainers priced at \(拢 57.74,\) a T-shirt priced at \(拢 17.28\), and a yo-yo priced at \(拢 4.98 .\) Using only mental arithmetic, work out how much I should expect to pay altogether. (b) Some retailers display prices without adding VAT - or "sales tax" - at \(20 \%\) (because their main customers need to know the pre-VAT price). Suppose the prices in part (a) are the prices before adding VAT. Each price then needs to be adjusted in two ways - adding VAT and subtracting the discount. Should I add the VAT first and then work out the discount? Or should I apply the discount first and then add the VAT? (c) Suppose the discount in part (b) is no longer \(15 \%\). What level of discount would exactly cancel out the addition of VAT at \(20 \% ?\)

Let \(\Delta=\operatorname{area}(\triangle A B C)\) (a) Prove that $$ \Delta=\frac{1}{2} \cdot a b \cdot \sin C . $$ (b) Prove that \(4 R \Delta=a b c\).
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