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 the total production in the first $7$ years.

Answer 1

Let production of $T.V$ sets in first years is $a$. Given that production of $T.V$ sets in third year
$a_3=600$
and in seventh year
$a_7=100$
Now, $a_3=a+(3-1)d$
$600=a+2d$.......(i)
and $a_7=a+(7-1)d$
$700=a+6d$.....(ii)

Subtracting equation (i) from (ii),
$100=4d$
$d=\frac{100}{5}=25$

Put the value of $d$ in equation (i)
$\Rightarrow 600=a+2\times 25$
$\Rightarrow 600=a+50$
$\Rightarrow a=600-50$
$\Rightarrow a=550$

Production of $T.V$ set in $1^{st}$ year $=550$

Total production in $7$ years can be calculated using sum of first 'n' terms of an AP formula.

$S_n=\frac{7}{2}[2a+(n-1\left)d\right]$

By formula,
$S_7=\frac{7}{2}[2\times 550+(7-1\left)25\right]$

$=\frac{7}{2}[1100+6\times 25]$

$=\frac{7}{2}[1100+150]$

$=\frac{7}{2}\times 1250=7\times 625$

$=4375$

Thus, total production in $7$ years in $4375$ sets.