arithmetic sequence 等差數列

Two functions have similar structure with particular \(a\)

Problem Statement: We are given the functions \[f(x)=e^x−ax\] and \[g(x)=ax−\ln x\] with \(x>0\) for \(g(x)\) It is given that \(f(x)\) and \(g(x)\) have the same minimum value. (1) Find \(a\) (2) Prove that there exists a line \(y=b\) intersecting both curves \(y=f(x)\) and \(y=g(x)\) at three distinct points whose \(x\)-coordinates form an arithmetic progression Answer: (1) \(f'(x)=e^x-a\) \(f′(x)=0\) ⟹ \(0=e^x-a\) ∴ \(x= \ln a\) The minimum value... » read more

Free Challenging Questions : M0-07-08

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-08 Consider a sequence with the first term \(a\) and the final term \(b\), # More challenging questions will be updated # Answer

Free Challenging Questions : M0-07-04

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-04 The first, third, and 13th terms of an arithmetic sequence form a geometric sequence. Find the common ratio of the geometric sequence. # More challenging questions will be updated # Answer

Free Challenging Questions : M0-07-03

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-03 In an arithmetic sequence, the sum of the first \(n\) terms is equal to the sum of the next \(n+1\) terms, where \(n \geq 1\). Which term in this sequence is equal to \(0\)? # More challenging questions... » read more

Free Challenging Questions : M0-07-02

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-02 If \(a^2\), \(b^2\) and \(c^2\) form an arithmetic sequence where \(a\), \(b\) and \(c\) are real numbers, show that \(\frac{1}{b+c}\), \(\frac{1}{c+a}\) and \(\frac{1}{a+b}\) also form an arithmetic sequence. # More challenging questions will be updated # Answer

Free Challenging Questions : M0-07-01

Challenging Questions Mathematics – Compulsory Part Learning Unit: Arithmetic and geometric sequences and their summations Question M0-07-01 Consider a sequence \(a, b, c, \ldots \ldots\) # More challenging questions will be updated # Answer