Home -> Solved problems -> Challenging problem

Challenging problem

Solution

Let’s start, we have

$$\begin{array}{rl}{a}_{n+1}& =2^n-3a_n=2^n-3\times 2^{n-1}+3^2{a}_{n-1}\\ & =\cdots =2^n+\sum _{j=1}^n(-3{)}^j{2}^{n-j}+(-3{)}^{n+1}{a}_0\\ & =\frac{2^{n+1}}{5}+(a_0-\frac{1}{5})(-3{)}^{n+1}\end{array}$$

If $a_0=\frac{1}{5}$, then we have no problem. If $a_0\ne \frac{1}{5}$, ${\left(\frac{2}{3}\right)}^n$ goes to 0 when $n\to \mathrm{\infty }$, so $\frac{a_n}{3^n}$ will have the same sign as $(a_0-\frac{1}{5}){(-1)}^n$ when $n$ is large. Hence, $a_n<a_{n+1}$ will not hold then contradiction

Home -> Solved problems -> Challenging problem

Related Topics

Find the volume of the square pyramid as a function of $a$ and $H$ by slicing method.

Solution

Prove that $$\underset{x\to 0}{lim}\frac{\sin x}{x}=1$$

Solution

Prove that

Solution

Calculate the half derivative of $x$

Solution

Prove Wallis Product Using Integration

Solution

Calculate the radius R

Solution

Calculate the volume of Torus using cylindrical shells

Solution

Find the derivative of exponential $x$ from first principles

Solution

Calculate the sum of areas of the three squares

Solution

Find the equation of the curve formed by a cable suspended between two points at the same height

Solution

Solve the equation for real values of $x$

Solution

Solve the equation for $xϵ\mathbb{R}$

Solution

Determine the angle $x$

Solution

Calculate the following limit

Solution

Calculate the following limit

Solution

Calculate the integral

Solution

Challenging problem

Solution

Home -> Solved problems -> Challenging problem

Share the solution: Challenging problem