The number of mobile telephones manufactured by a new company increase

1.	The number of mobile telephones manufactured by a new company increasesevery month by a constant number. If the company manufactures 600 mobiles in the 3rd month and 1600 mobiles in the 7th month, find the total product in the first year.
Answer» AnswEr : If Company is increasing manufacturing of MOBILES by a constant number every month that means it is ARITHMETIC Progresion. Let the Mobile PRODUCED in FIRST Month be a and, Constant Increase i.e. COMMON Difference be d. $\bigstar\:\boxed{\sf T_n =a + (n - 1)d}$ $\rule{150}{1}$ $\underline{\bf{\dag}\:\textit{Phone Produced in 3rd Month :}}$ $\longrightarrow\tt T_n = a +(n - 1)d\\\\\\\longrightarrow\tt T_3 = a + (3-1)d\\\\\\\longrightarrow\tt 600 = a + 2d\qquad\dfrac{\qquad}{}eq.(1)$ $\rule{120}{2}$ $\underline{\bf{\dag}\:\textit{Phone Produced in 7th Month :}}$ $\longrightarrow\tt T_n = a +(n - 1)d\\\\\\\longrightarrow\tt T_7 = a + (7-1)d\\\\\\\longrightarrow\tt 1600 = a + 6d\qquad\dfrac{\qquad}{}eq.(2)$ $\rule{200}{1}$ $\underline{\bigstar\:\textsf{Subtracting eq.(1) from eq.(2) :}}\\\\:\implies\tt 1600 = a + 6d\\\\:\implies\tt 600 = a + 2d\\\dfrac{\qquad\qquad\qquad\qquad}{}\\:\implies\tt (1600-600) = (6d-2d) \\\\\\:\implies\tt 1000 = 4d\\\\\\:\implies\tt\dfrac{1000}{4} = d\\\\\\:\implies \boxed{\tt d = 250} \\\\\\\underline{\bigstar\:\textsf{Putting the value of d in eq.(1) :}}\\\\:\implies\tt600 = a + 2d\\\\\\:\implies\tt600 = a + 2(250)\\\\\\:\implies\tt600 = a + 500\\\\\\:\implies\tt600 - 500 = a\\\\\\:\implies\boxed{\tt a = 100}$ $\rule{250}{2}$ $\underline{\bf{\dag}\:\textit{Phone Produced in First Year :}}\\\\: \implies\tt S_n =\dfrac{n}{2} \bigg [2a + (n - 1)d \bigg]\\\\\\:\implies \tt S_{12} =\dfrac{12}{2}\bigg [2(100) + (12 - 1)250\bigg ]\\\\\\:\implies \tt S_{12} =6[200 + (11 \times 250)]\\\\\\:\implies \tt S_{12} = 6[200 + 2750]\\\\\\:\implies \tt S_{12} = 6 \times 2950\\\\\\:\implies \large\boxed{\tt S_{12} =17700}$ ⠀ $\therefore\:\underline{\textsf{Total Phone Produced in First Year is \textbf{17700.}}}$

The number of mobile telephones manufactured by a new company increasesevery month by a constant number. If the company manufactures 600 mobiles in the 3rd month and 1600 mobiles in the 7th month, find the total product in the first year.

Answer»

AnswEr :

If Company is increasing manufacturing of MOBILES by a constant number every month that means it is ARITHMETIC Progresion.

Let the Mobile PRODUCED in FIRST Month be a and, Constant Increase i.e. COMMON Difference be d.

$\bigstar\:\boxed{\sf T_n =a + (n - 1)d}$

$\rule{150}{1}$

$\underline{\bf{\dag}\:\textit{Phone Produced in 3rd Month :}}$

$\longrightarrow\tt T_n = a +(n - 1)d\\\\\\\longrightarrow\tt T_3 = a + (3-1)d\\\\\\\longrightarrow\tt 600 = a + 2d\qquad\dfrac{\qquad}{}eq.(1)$

$\rule{120}{2}$

$\underline{\bf{\dag}\:\textit{Phone Produced in 7th Month :}}$

$\longrightarrow\tt T_n = a +(n - 1)d\\\\\\\longrightarrow\tt T_7 = a + (7-1)d\\\\\\\longrightarrow\tt 1600 = a + 6d\qquad\dfrac{\qquad}{}eq.(2)$

$\rule{200}{1}$

$\underline{\bigstar\:\textsf{Subtracting eq.(1) from eq.(2) :}}\\\\:\implies\tt 1600 = a + 6d\\\\:\implies\tt 600 = a + 2d\\\dfrac{\qquad\qquad\qquad\qquad}{}\\:\implies\tt (1600-600) = (6d-2d) \\\\\\:\implies\tt 1000 = 4d\\\\\\:\implies\tt\dfrac{1000}{4} = d\\\\\\:\implies \boxed{\tt d = 250} \\\\\\\underline{\bigstar\:\textsf{Putting the value of d in eq.(1) :}}\\\\:\implies\tt600 = a + 2d\\\\\\:\implies\tt600 = a + 2(250)\\\\\\:\implies\tt600 = a + 500\\\\\\:\implies\tt600 - 500 = a\\\\\\:\implies\boxed{\tt a = 100}$

$\rule{250}{2}$

$\underline{\bf{\dag}\:\textit{Phone Produced in First Year :}}\\\\: \implies\tt S_n =\dfrac{n}{2} \bigg [2a + (n - 1)d \bigg]\\\\\\:\implies \tt S_{12} =\dfrac{12}{2}\bigg [2(100) + (12 - 1)250\bigg ]\\\\\\:\implies \tt S_{12} =6[200 + (11 \times 250)]\\\\\\:\implies \tt S_{12} = 6[200 + 2750]\\\\\\:\implies \tt S_{12} = 6 \times 2950\\\\\\:\implies \large\boxed{\tt S_{12} =17700}$

⠀

$\therefore\:\underline{\textsf{Total Phone Produced in First Year is \textbf{17700.}}}$