Question

A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find :
(i) the production in the $1^{st}$ year
(ii) the production in the ${10}^{th}$ year
(iii) the total production in first $7$ years

Answer 1

(i)

Since the production increases uniformly by a fixed number every year,

the number of TV sets manufactured in $1^{st}$,$2^{nd}$ , $3^{rd}$ 3rd, . . ., years will form an AP.

Let us denote the number of TV sets manufactured in the nth year by an.

Then, $a_3=600and{a}_7=700$

$\Rightarrow a+2d=600;anda+6d=700$

Solving these equations, we get $d=25$ and $a=550$.

Therefore, production of TV sets in the first year is $550$.

(ii)

Now $a_{10}=a+9d=550+9\times 25=775$

So, production of TV sets in the 10th year is $775$.

(iii)

Also, $S_7=\frac{7}{2}\left(2\right(550)+(7-1\left)25\right)$

$=\left(\frac{7}{2}\right)[1100+150]$

$=4375$

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