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## Numbers and Algebra

### Advanced Algebra - Exponential and Logarithmic Functions

#### Q.03

'(3) (2) from $M_{n}=\\frac{1}{n} \\sum_{k=1}^{n} c_{k}=\\frac{1}{n} \\cdot \\frac{n}{2}(2 \\log _{2} a+\\frac{n-1}{2} \\log _{2} r)$\n\n$=\\log _{2} a+\\frac{n-1}{4} \\log _{2} r=\\log _{2} a r^{\\frac{n-1}{4}}$\n\nTherefore, from $d_{n}=2^{M_{n}}$ we have d_{n}=2^{\\log _{2} a r^{\\frac{n-1}{4}}}=\\operatorname{\overline} \\frac{n-1}{4}\nHence $\\frac{d_{n+1}}{d_{n}}=\\frac{a r^{\\frac{n}{4}}}{a r^{\\frac{n-1}{4}}}=r^{\\frac{1}{4}}$ (constant).\nTherefore, the sequence $\\left\\{d_{n}\\right\\}$ is a geometric sequence with first term $d_{1}=a$ and common ratio $r^{\\frac{1}{4}}$.'

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'What were the achievements of John Napier (1550-1617)?'

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'(1) Let $\\log _{2} a=\\log _{3} b=k$, then $a>1, b>1$\n\\[\egin{array}{l}\nk>0 \\quad \\text { and } a=2^{k}, b=3^{k} \\\\\n\\text { Now } \\quad\\left(a^{\\frac{1}{2}}\\right)^{6}-\\left(b^{\\frac{1}{3}}\\right)^{6}=a^{3}-b^{2}=\\left(2^{k}\\right)^{3}-\\left(3^{k}\\right)^{2}=8^{k}-9^{k}<0 \\\\\n\\text { Therefore } \\quad\\left(a^{\\frac{1}{2}}\\right)^{6}<\\left(b^{\\frac{1}{3}}\\right)^{6} \\\\\na>1, \\quad b>1 \\text { so } \\quad a^{\\frac{1}{2}}<b^{\\frac{1}{3}} \\\\\n\\end{array}\\]'

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'Common Logarithm Table: Table of logarithms with base 10.'

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'Find the sum of the following series. Given n≧2:\n(1) 1•2^{3} + 2•2^{4} + 3•2^{5} + ... + n•2^{n+2}'

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'Given that the sum of the first 8 terms of a geometric sequence is 54, and the sum of the first 16 terms is 63, find the sum of terms 17 to 24 of this geometric sequence.'

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'65 (1) 1.5 < \\log _{4} 9 < \\log _{2} 5\n(2) \\log _{4} 2 < \\log _{3} 4 < \\log _{2} 3'

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'Practice problem: Let log_{2} x=t, where 1≤x≤8 corresponds to 0≤t≤3. Also, log_{1/2} x=-log_{2} x=-t. Define y=t^{2}-2 t+3 as a function of t. Find the maximum and minimum values of y within the range 0≤t≤3.'

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'Chapter 7 Exponential and Logarithmic Functions-147'

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'If \ \\log_{3} 2=a, \\log_{5} 4=b \, express \ \\log_{15} 8 \ in terms of \ a \ and \ b \.'

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'When you want to learn advanced topics, which pages should you refer to?'

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'Prove that if 16^4 * x + y + z = 1 / x + 1 / y + 1 / z = 1, then at least one of x, y, or z must be 1.'

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'Confirmation of conditions for logarithmic equations and real numbers'

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'Find the general term of the recurrence relation $a_{n+1}=3a_n+2n-1$.'

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'If you deposit 1 million yen with an annual interest rate of 1% compounded annually, in how many years will the total amount first exceed 1.1 million yen? It is permissible to use common logarithm table.'

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'Here are two examples where an infinite geometric series is used: 1. Trisection of a square Divide a square paper with area 1 into four equal parts in a cross shape, and distribute one each to A, B, and C. Divide the remaining one into four equal parts again, and distribute one each to A, B, and C. Repeat this process infinitely, the total area of paper received by A, B, and C can be expressed as the following infinite geometric series ∑(1/4)^n (from n=1 to ∞). Find the sum of this infinite geometric series.'

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'In order for the sequence $\\left\\{\\left(\\frac{5x}{x^{2}+6}\\right)^{n}\\right\\}$ to converge, determine the range of real numbers for $x$. Also, find the limit of the sequence at that time.'

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'(1) Eliminate A, B from the equation y=A \\sin x + B \\cos x -1 to obtain the differential equation ③ 213.'

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'For a ball launched straight up at a certain speed, let h meters be the height above the ground x seconds after launch. When the value of h is given by h=-5x²+40x, in what range of x values is the ball at a height between 35m and 65m from the ground?'

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'Let f(x) = (log_{2} \x0crac{x}{a})(log_{2} \x0crac{x}{b}) (where a b = 8, a > b > 0). If the minimum value of f(x) is -1, find the value of a^2.'

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'For a sequence \ \\left\\{a_{n}\\right\\} \, it is assumed that the sum from the initial term \ p a_{1} \ to the nth term \ p^{n} a_{n} \ of the sequence \ \\left\\{p^{n} a_{n}\\right\\} \ is equal to \ q^{n} \. Where, \ p \\neq 0 \. \n(1) Find \ a_{n} \. \n(2) Find \ S_{n}=a_{1}+a_{2}+\\cdots+a_{n} \.'

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"(3) Let y=x^{3}+4 x^{2}+6 x-1, then y'=3 x^{2}+8 x+6=3(x+4/3)^{2}+2/3 is greater than 0 for all real numbers, which means y is increasing. Therefore, the equation x^{3}+4 x^{2}+6 x-1=0 has 1 real root."

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'(2) Let \ \\log _{3} 7=a, \\log _{4} 7=b \. Find \ \\log _{12} 7 \ in terms of \a, b\.'

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'Solve the following equations and inequalities, where a is a positive constant not equal to 1.'

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'Arrange the values log_{a} b, log_{b} a, log_{a}(\\frac{a}{b}), log_{b}(\\frac{b}{a}), 0, \\frac{1}{2}, 1 in ascending order when 1 < a < b < a^{2}.'

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'Answer the following questions about the properties of logarithmic functions.'

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'Consider the scale of the logarithmic scale shown below.'

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'For the complex number z, the function e^z is defined by replacing 11 with x in the expression'

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"Practice 67 |II| Book p.558 (1) f'(x) = (1 + x/√(1+x^2)) / (x + √(1+x^2)) = 1/√(1+x^2) (2) Polar equation r=θ(θ≧0) gives x=r cosθ = θ cosθ, y=r sinθ = θ sinθ where dx/dθ = cosθ − θ sinθ, dy/dθ = sinθ + θ cosθ Therefore, the table of increasing and decreasing values of x, y with respect to θ is as follows. θ = 0 ... α ... β ... π dx/dθ + 0 - - - x ↗ local max ↘ ↘ dy/dθ + + + 0 - y ↗ ↗ local max ↘ However, \\cos α−α\\sin α=0 is the verification condition \\sin β+β\\cos β=0"

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'Using the intermediate value theorem\n(1) Prove that the equation \\( 3^{x}=2(x+1) \\) has at least one real solution in the range \ 1<x<2 \.\n(2) Let \\( f(x), g(x) \\) be continuous functions on the interval \ [a, b] \. If \\( f(a)>g(a) \\) and \\( f(b)<g(b) \\), show that the equation \\( f(x)=g(x) \\) has at least one real solution in the range \ a<x<b \.'

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'Please translate the given text into multiple languages.'

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'In Chapter 2, let there be constants a, b such that 100<a<b. Define x_n=( (a^n/b + b^n/a)^(1/n) ) (n=1,2,3,...). Find (1) Prove the inequality b^n < a(x_n)^n < 2b^n. (2) Find the limit lim n->∞ x_n.'

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'Given equation 120(3) \\( \\left(\\log _{2} \\frac{x}{a}\\right)\\left(\\log _{2} \\frac{x}{b}\\right) \\left(ab=8, \\quad a=3, x=0\\right)\\)'

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'Express the size of each set of numbers using inequality symbols.'

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'(2) Simplify the following expressions:\n(a) \ \\log _{0.2} 125 \\n(b) \ \\log _{6} 12+\\log _{6} 3 \\n(c) \ \\log _{3} 18-\\log _{3} 2 \\n(d) \ 6 \\log _{2} \\sqrt[3]{10}-2 \\log _{2} 5 \\n(e) \ \\frac{1}{2} \\log _{10} \\frac{5}{6}+\\log _{10} \\sqrt{7.5}+\\frac{1}{2} \\log _{10} 1.6 \'

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'Values of expressions involving both exponential and logarithmic functions'

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'Limits of Sequences (5) ... using the squeeze theorem and binomial theorem'

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'Let fn(x) = (log x)^n (where n is an integer greater than or equal to 3). Here, log x is the natural logarithm. Find the values of n and x_0 when the curve y = fn(x) has a point of inflection (x_0, 8), and sketch the general shape of the curve (including concavity). [Job Development University]'

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'Prove that the equation 3^x=2(x+1) has at least one real solution in the range of 1<x<2.'

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'Practice let n be a natural number greater than or equal to 2.'

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'Let n be a natural number. Show that the nth derivative f^{(n)}(x) of the function f(x)=x^{2} e^{x} can be expressed as f^{(n)}(x)=x^{2} e^{x}+2 n x e^{x}+a_{n} e^{x}, where a_{n} is a constant, and find the value of a_{n}.'

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'Find the values of the constants a and b such that y=e^{3x}(a \\sin 2x+b \\cos 2x) and y^{\\prime}=e^{3x} \\sin 2x hold true.'

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'Throw n balls into 2n boxes. Assume each ball will be placed in one of the boxes with equal probability. Let p_{n} be the probability that each box contains at most 1 ball. Find the limit \ \\lim _{n \\rightarrow \\infty} \\frac{\\log p_{n}}{n} \.'

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'For any real numbers c_{1}, c_{2}, the function f(x)=c_{1} e^{2x}+c_{2} e^{5x} satisfies the relationship f’’(x) − a f’(x)+b f(x)=0. [Keio University]'

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"Please provide the page containing 'Euler's formula'."

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'Calculate the number of digits of 3^n for a natural number n and find its limit.'

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'Given constants \ a, b \ where \ 0 < a < b \. Let \\( x_{n}=\\left(\\frac{a^{n}}{b}+\\frac{b^{n}}{a}\\right)^{\\frac{1}{n}} \\), prove (1) the inequality \\( b^{n} < a\\left(x_{n}\\right)^{n} < 2b^{n} \\). (2) Find \ \\lim _{n \\rightarrow \\infty} x_{n} \.'

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'Translate the given text of problem 309 in mathematics from Japanese'

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'21 (1) \\( b_n = -(-3)^{n-1} \\)\n(2) \\( a_n=\\frac{3(-3)^{n-1}+1}{(-3)^{n-1}+1}, \\lim _{n \\rightarrow \\infty} a_{n}=3 \\)\n'

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'A point P moving on a number line is given by the velocity v at time t as v=t^{3}, and at t=0, P is at the origin. Find: \n(1) The coordinate x of P at t=2. \n(2) The distance s traveled by P from t=0 to t=2.'

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'(1) Let a be a non-zero constant. For x≥0, find f(x)=lim(n→∞)(x^(2n+1)+(a-1)x^n-1)/(x^(2n)-ax^n-1).'

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'Investigate the convergence and divergence of the following infinite geometric series, and find the sum if it converges.'

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'Prove with mathematical induction that for any natural number n, the following inequality holds true for x>0: e^x > 1 + x + x^2/2! + x^3/3! + ... + x^n/n!'

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'(3) \\frac{1}{2} \\log \\frac{4 e(e+2)}{3(e+1)^{2}}'

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'16\n(3)\n\\[\n\egin{array}{l} \ny^{\\prime}=e^{3 x} \\cdot(3 x)^{\\prime}=3 e^{3 x} \\\\\ny^{\\prime \\prime}=3 e^{3 x} \\cdot(3 x)^{\\prime}=9 e^{3 x} \\\\\n\\text { Therefore } \\quad y^{\\prime \\prime \\prime}=9 e^{3 x} \\cdot(3 x)^{\\prime}=27 e^{3 x}\n\\end{array}\n\\]'

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'Prove that the equation 3^x = 2(x+1) has at least one real solution in the range 1<x<2.'

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'Create a PR container. Pour water gently into this container at a rate of a per unit time. Let V represent the volume of water when the water height is h, the water radius is r, the water area is S, and the water volume is V after time t since pouring started. (1) Express V. (2) Express the rates of change dh/dt, dr/dt, dS/dt of h, r, S with respect to time t using a and h.'