QuestionANS: $x \in(1,5) \cup 18 ;+$
3. If $S(N)=\sum_{k=1}^{N} k$ then which value of $N$ solves the following equation? $$\sum_{n=1}^{S(N)} 4^{n}=\frac{4}{3}\left(4^{55}-1\right)$$
ANS: $N=10$

Studdy Solution
Substitute $S(N) = 55$ into the expression for $S(N)$:
$$\frac{N(N+1)}{2} = 55$$
Multiply both sides by 2:
$$N(N+1) = 110$$
Solve the quadratic equation:
$$N^2 + N - 110 = 0$$
Using the quadratic formula $N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 1$, and $c = -110$:
$$N = \frac{-1 \pm \sqrt{1 + 440}}{2}$$ $$N = \frac{-1 \pm \sqrt{441}}{2}$$ $$N = \frac{-1 \pm 21}{2}$$
The positive solution is:
$$N = \frac{20}{2} = 10$$
The value of $N$ is:
$$\boxed{10}$$