Arithmetic Progression

Find the sum of the terms from the (n + 1) th to the m th term inclusive of an arithmetical progression whose

first term is a and whose second term is b.

If m = 13, n =   3 and the sum is 12a, find the ratio b: a.

Answer (m-n){a+1/2(m+n-1)(b-a)} ; 77/75

$\text{Formula arithmetical progression: }\\ \begin{array}{|rcll|} \hline a_1 &=& a \\ a_2 &=& a+d \\ \hline \end{array}$

$\text{second term is b: }\\ \begin{array}{|rcll|} \hline a_2 &=& a+d \\ a_2 &=& b \\ \hline a+d &=& b \\ \mathbf{d} & \mathbf{=} & \mathbf{b-a} \\ \hline \end{array}$

The common difference is $\mathbf{b-a}$

$\text{The sum of the member is: }\\ \begin{array}{|rcll|} \hline s_n = n\cdot a + \dfrac{n(n-1)}{2}\cdot d \\ \hline \end{array}$

$\text{The sum of the terms from the (n + 1) th to the m th term is: }\\ \begin{array}{|rcll|} \hline s_m-s_n &=& \underbrace{m\cdot a + \dfrac{m(m-1)}{2}\cdot (b-a)}_{=s_m} - \left[ \underbrace{n\cdot a + \dfrac{n(n-1)}{2}\cdot (b-a)}_{=s_n} \right] \\\\ &=& a(m-n) + \left(\dfrac{b-a}{2}\right)[ m(m-1)-n(n-1) ] \\\\ &=& a(m-n) + \left(\dfrac{b-a}{2}\right)( m^2-m-n^2+n ) \\\\ &=& a(m-n) + \left(\dfrac{b-a}{2}\right)( m^2-n^2-m+n ) \\\\ &=& a(m-n) + \left(\dfrac{b-a}{2}\right)[ m^2-n^2-(m-n) ] \\\\ &=& a(m-n) + \left(\dfrac{b-a}{2}\right)[ (m-n)(m+n)-(m-n) ] \\\\ &=& a(m-n) + \left(\dfrac{b-a}{2}\right)(m-n)( m+n-1 ) \\\\ \mathbf{s_m-s_n} &\mathbf{=}& \mathbf{ (m-n)\left[ a + \left(\dfrac{b-a}{2}\right)( m+n-1 ) \right] } \\ \hline \end{array}$

$\small{ \text{If $m = 13, n = 3$ and the sum is $12a$, find the ratio $b: a$ }\\ \begin{array}{|rcll|} \hline \mathbf{s_m-s_n} &\mathbf{=}& \mathbf{ (m-n)\left[ a + \left(\dfrac{b-a}{2}\right)( m+n-1 ) \right] } \\\\ 12a & = & (m-n)\left[ a + \left(\dfrac{b-a}{2}\right)( m+n-1 ) \right] \\\\ \dfrac{12a}{m-n} & = & a + \left(\dfrac{b-a}{2}\right)( m+n-1 ) \\\\ \dfrac{12a}{m-n} & = & a + \left(\dfrac{b}{2}\right)( m+n-1 )- \left(\dfrac{a}{2}\right)( m+n-1 ) \\\\ a\left( \dfrac{12}{m-n} - 1+ \dfrac{m+n-1}{2} \right) & = & b\left( \dfrac{m+n-1}{2} \right) \qquad m = 13 \qquad n = 3 \\\\ a\left( \dfrac{12}{10} - 1+ \dfrac{15}{2} \right) & = & b\left( \dfrac{15}{2} \right) \\\\ \dfrac{b}{a} &=& \left( \dfrac{2}{15} \right)\left( \dfrac{12}{10} - 1+ \dfrac{15}{2} \right) \\\\ &=& \left( \dfrac{2}{15} \right)\left( \dfrac{12-10+75}{10} \right) \\\\ &=& \dfrac{2\cdot 77}{150} \\\\ &=& \dfrac{2\cdot 77}{2\cdot 75} \\\\ \mathbf{\dfrac{b}{a}} &\mathbf{ =}& \mathbf{ \dfrac{77}{75}} \\ \hline \end{array} }$

12a = (m-n){a+1/2(m+n-1)(b-a)} , where m=13, n = 3

12a =(13 - 3)* [a + 1/2(13+3 -1)*(b -a)]

12a = 10*[a + 1/2(15)*(b - a)]

12a = 10*[a + 1/2(15b - 15a)]

12a = 10a + 5(15b - 15a)

12a = 10a + 75b - 75a

12a - 10a + 75a = 75b

77a = 75b

b =77a / 75, or

b/a = 77 / 75