Question

A manufacture of TV sets produced $600$ units in the third year and $700$ units in the seventh year. Assuming that the production increase uniformly by a fixed number every year, find (i) the production in ${10}^{th}$ year (ii) the total production in the first $7$ years.

Answer 1

Since the production increases uniformly by a fixed number every year. Therefore, the sequence formed by the production in different years is an A.P.

Let $a$ be the first year and $d$ be the common difference of the A.P. formed i.e. '$a$' denote the production in the first year and $d$ denotes the number of units by which the production increases every year.

We have,
$a_3=600$ and $a_7=700$
$⟹a+2d=600$ ........ (1)

and $a+6d=700$ ......... (2)
Solving these equation, we get
$a=550$ and $d=25$

(i) We have,
Production in the ${10}^{th}$ year $=a_{10}=a+9d=550+9\times 25=775$
So, production in ${10}^{th}$ year is of $775$ TV sets.
(ii) We have,
Total production in $7$ years
$=$ Sum of $7$ terms of the A.P. with first term $a(=550)$ and common difference $d(=25)$.
$=\frac{7}{2}\{2\times 550+(7-1)\times 25\}$
$=\frac{7}{2}(1100+150)=4375$

Thus, the total production in $7$ years is of $4375$ Tv sets.