The number \(210\) can be written as the sum of consecutive integers in several ways.
(a) When Written as the sum of the greatest possible number of consecutive positive integers, what is the largest of these integers?
(b) What if we allow negative integerrs into the sum; then what is the greatest possible number of consecutive integers that sum to \(210\)?
Arithmetic progression:
\(\begin{array}{|rcll|} \hline a_n &=& a_1+(n-1)*d \quad | \quad d=1 \\ a_n &=& a_1+(n-1)*1 \\ a_n &=& a_1+n-1 \\ \mathbf{n} &=& \mathbf{a_n-a_1+1} \\ \hline \end{array}\)
Sum of the Arithmetic progression:
\(\begin{array}{|rcll|} \hline \text{sum} &=& \dfrac{a_1+a_n}{2}*n \quad &|\quad \text{sum} = 210 \\\\ 210 &=& \dfrac{a_1+a_n}{2}*n \\\\ \left(\dfrac{a_1+a_n}{2}\right)*n &=& 210 \\\\ \mathbf{(a_1+a_n)*n} &=& \mathbf{420} \\ \hline \end{array}\)
The divisors of \(420\) are:
Divisors:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 10 |
12 | 14 | 15 | 20 | 21 | 28 | 30 | 35 |
42 | 60 | 70 | 84 | 105 | 140 | 210 | 420 (24 divisors)
\(\text{Let $a_1+a_n=k\quad$or$\quad \mathbf{a_n=k-a_1}$} \)
\(\begin{array}{|rcll|} \hline (a_1+a_n)*n &=& 420 \\ k*n &=& 1*420 \\ &=& 2*210 \\ &=& 3*140 \\ &=& 4*105 \\ &=& 5*84 \\ &=& 6*70 \\ &=& 7*60 \\ &=& 10*42 \\ &=& 12*35 \\ &=& 14*30 \\ &=& 15*28 \\ &=& 20*21 \\ &=& 21*20 \\ &=& 28*15 \\ &=& 30*14 \\ &=& 35*12 \\ &=& 42*10 \\ &=& 60*7 \\ &=& 70*6 \\ &=& 84*5 \\ &=& 105*4 \\ &=& 140*3 \\ &=& 210*2 \\ &=& 420*1 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline n &=& a_n-a_1+1 \quad &| \quad a_n = k - a_1 \\ n &=& k-a_1-a_1+1 \\ n &=& k-2a_1+1 \\ 2a_1 &=& 1+k-n \\\\ \mathbf{a_1} &=& \mathbf{ \dfrac{1+k-n}{2} } \\ \hline \mathbf{a_n} &=& \mathbf{ k-a_1 } \\ \hline \end{array}\)
\(\begin{array}{|r|r|r|r|r|l|} \hline k*n & k & n & a_1 & a_n & \text{Arithmetic} \\ &&&& & \text{progression} \\ \hline 1*420 & 1 &\color{red}420 & -209 & 210 &-209,-208,\dots,209,210 \\ 2*210 & 2 & 210 & (-103.5)& &\text{$a_1$ no integer } \\ 3*140 & 3 & 140 & -68 & 71 &-68,-67,\dots,70,71 \\ 4*105 & 4 & 105 & -50 & 54 &-50,-49,\dots,53,54 \\ 5*84 & 5 & 84 & -39 & 44 &-39,-38,\dots,43,44 \\ 6*70 & 6 & 70 & (-31.5)& &\text{$a_1$ no integer } \\ 7*60 & 7 & 60 & -26 & 33 &-26,-25,\dots,32,33 \\ 10*42 & 10 & 42 & (-15.5)& &\text{$a_1$ no integer } \\ 12*35 & 12 & 35 & -11 & 23 &-11,-10,\dots,22,23\\ 14*30 & 14 & 30 & (-7.5)& &\text{$a_1$ no integer } \\ 15*28 & 15 & 28 & -6 & 21 &-6,-5,\dots,20,21\\ \hline 20*21 & 20 & 21 & (0) & 20 &\text{$a_1$ no integer } \\ \hline 21*20 & 21 & \color{red}20 & 1 & 20 & 1,2,\dots,19,20 \\ 28*15 & 28 & 15 & 7 & 21 & 7,8,\dots,20,21\\ 30*14 & 30 & 14 & (8.5)& &\text{$a_1$ no integer } \\ 35*12 & 35 & 12 & 12 & 23 & 12,13,\dots,22,23\\ 42*10 & 42 & 10 & (16.5)& &\text{$a_1$ no integer } \\ 60*7 & 60 & 7 & 27 & 33 & 27,28,\dots,32,33\\ 70*6 & 70 & 6 & (32.5)& &\text{$a_1$ no integer } \\ 84*5 & 84 & 5 & 40 & 44 & 40,41,42,43,44 \\ 105*4 &105 & 4 & 51 & 54 & 51,52,53,54\\ 140*3 &140 & 3 & 69 & 71 & 69,70,71\\ 210*2 &210 & 2 & (104.5)& &\text{$a_1$ no integer } \\ 420*1 &420 & 1 & 210 & 210 & 210 \\ \hline \end{array}\)
(a) 20
(b) 420
+118654 
The answers given are correct, but here is one way we can get those answers. We are asked to find the perimeter of the triangle ADE. We know DE has length 5, so we must find the lengths of the remaining two sides.
