Part (b): Find all pairs of positive integers (a, n) such that n \ge 2 and
\[a + (a + 1) + (a + 2) + \dots + (a + n - 1) = 100.\]
+26367 Part (b):
Find all pairs of positive integers \((a, n)\) such that \(n \ge 2\) and
\(a + (a + 1) + (a + 2) + \ldots + (a + n - 1) = 100\).
\(\begin{array}{|rcll|} \hline a + (a + 1) + (a + 2) + \dots + (a + n - 1) &=& 100 \\\\ \underbrace{(a + a + a + \ldots + a)}^{n ~\text{times}}_{\text{sum}=n\cdot a} + \underbrace{( 1 + 2 + \ldots + n - 1)}_{\text{sum}=\frac{1+(n-1)}{2} \cdot(n-1) }&=& 100 \\\\ n\cdot a + \dfrac{1+(n-1)}{2} \cdot(n-1) &=& 100 \\\\ n\cdot a + \dfrac{n(n-1)}{2} &=& 100 \quad & | \quad \cdot 2 \\\\ 2n\cdot a + n(n-1) &=& 200 \\ \mathbf{n\cdot (\underbrace{2a+n-1)}_{=b}} &\mathbf{=}& \mathbf{\underbrace{200}_{=n\cdot b}} \\ \hline \end{array} \)
Divisors of 200:
\(\begin{array}{|rcll|} \hline 200 = n &\cdot& b = \\ 1 &\cdot &200 \\ 2 &\cdot &100 \\ 4 &\cdot &50 \\ 5 &\cdot &40 \\ 8 &\cdot &25 \\ 10& \cdot & 20 \\ \hline \end{array} \)
\(\mathbf{(a,n) =\ ?}\)
\(\begin{array}{|r|r|r|c|} \hline n \ge 2 & b=2a+n-1 & a = \frac{b-n+1}{2} & a ~ \text{is integer} & (a,n) \\ \hline 2 & 100 \\ \hline 4 & 50 \\ \hline \color{red}5 & 40 & \color{red}18 & \checkmark & (18,5) \\ \hline \color{red}8 & 25 & \color{red}9 & \checkmark & (9,8) \\ \hline 10 & 20 \\ \hline \end{array} \)
\(\begin{array}{llcr} (18,5) :~ 18 + (18+1)+ (18+2)+ (18+3)+ (18+4)&=&100 \\ (9,8) :~ 9 + (9+1)+ (9+2)+ (9+3)+ (9+4)+ (9+5)+ (9+6)+ (9+7)&=&100 \\ \end{array}\)
