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प्रश्न
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.
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उत्तर
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 = \[a_{10}\]
\[a_{10} = a + \left( 10 - 1 \right)d\]
\[ = 550 + 9\left( 25 \right)\]
\[ = 775\]
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