Question

A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.

Answer 1

Let 

\[a_n\] denote the production of radio sets in the nth year.
Here,

\[a_3\] = 600,

\[a_7\] = 700
We know:

\[a_n = a + \left( n - 1 \right)d\]

\[a_3 = a + 2d\]

\[ \Rightarrow 600 = a + 2d . . . . . \left( 1 \right)\]

\[\text { And, } a_7 = a + 6d\]

\[ \Rightarrow 700 = a + 6d . . . . . \left( 2 \right)\]

Solving \[\left( 1 \right)\] and  \[\left( 2 \right)\] ,we get:
d = 25, a = 550
Hence, the production in the first year is 550 units.

(ii)  Let

\[S_n\] denote the total production in n years.
Total production in 7 years = \[S_7\]

\[= \frac{7}{2}\left\{ 2 \times 550 + \left( 7 - 1 \right)25 \right\}\]

\[ = 4375 \text { units }\]

(iii)  Production in the 10th year = \[a_{10}\]

                                      \[a_{10} = a + \left( 10 - 1 \right)d\]

                                              \[ = 550 + 9\left( 25 \right)\]

                                              \[ = 775\]