<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … is such that ${u_1 = 3}$ and ${{u}_{n+1}}=\frac{3}{4}{{u}_{n}}+4$, for ${ u_{n+1} = \frac{3}{4} u_n + 4, \quad \mathrm{for } n \geq 1}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express ${u_n}$ in the form ${16 + A\left(\frac{3}{4}\right)^{n-1}}$, where ${A}$ is a constant to be determined.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine whether the sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … converges.</span></div></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … is such that ${u_1 = 3}$ and ${{u}_{n+1}}=\frac{3}{4}{{u}_{n}}+4$, for ${ u_{n+1} = \frac{3}{4} u_n + 4, \quad \mathrm{for } n \geq 1}$.

Express ${u_n}$ in the form ${16 + A\left(\frac{3}{4}\right)^{n-1}}$, where ${A}$ is a constant to be determined.

Determine whether the sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … converges.

2009 IJC Promo Q8 (Modified)

2009 IJC Promo Q8 (Modified)

Video Solution: